Curto, Raúl E.; Gazeau, Jean-Pierre; Horzela, Andrzej; Moslehian, Mohammad Sal; Putinar, Mihai; Schmüdgen, Konrad; de Snoo, Henk; Stochel, Jan

Advances in Operator Theory
, Volume 5 (3) – Jul 8, 2020

/lp/springer-journals/mathematical-work-of-franciszek-hugon-szafraniec-and-its-impacts-GnXLOM8lBa

- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s) 2020
- ISSN
- 2662-2009
- eISSN
- 2538-225X
- DOI
- 10.1007/s43036-020-00089-z
- Publisher site
- See Article on Publisher Site

In this essay, we present an overview of some important mathematical works of Professor Franciszek Hugon Szafraniec and a survey of his achievements and inﬂuence. Keywords Szafraniec Mathematical work Biography Mathematics Subject Classiﬁcation 01A60 01A61 46-03 47-03 1 Biography Professor Franciszek Hugon Szafraniec’s mathematical career began in 1957 when he left his homeland Upper Silesia for Krakow to enter the Jagiellonian University. At that time he was 17 years old and, surprisingly, mathematics was his last-minute choice. However random this decision may have been, it was a fortunate one: he succeeded in achieving all the academic degrees up to the scientiﬁc title of professor in 1980. It turned out his choice to join the university shaped the Krakow mathematical community. Communicated by Qingxiang Xu. & Jan Stochel Jan.Stochel@im.uj.edu.pl Extended author information available on the last page of the article 1298 R. E. Curto et al. Professor Franciszek H. Szafraniec ´ ´ Krakow beyond Warsaw and Lwow belonged to the famous Polish School of Mathematics in the prewar period. Krako ´ w was a well known centre of Analysis therefore the School of Differential Equations ﬁtted with it. The main person of the school was Tadeusz Wazewski _ , who brought from the Paris school topological methods into the subject. Szafraniec was the last PhD student of Wazew _ ski beneﬁting from his ability to gather brilliant people around himself, directing them to interesting questions and, on the other hand encouraging to break mathematical boundaries. On this ground different members of Wazewski’s _ group spread over diverse areas of mathematics and so did Szafraniec. In 1968 he got converted to Operator Theory by Włodzimierz Mlak and soon after the passion which both of them had for this branch of mathematics was shared by their students and passed on to the next generations of mathematicians. This way Krakow became a vital centre of modern operator theory. The co-workers and former students of Szafraniec are present in all major Krakow universities. Thematic diversity along with his in-depth insight at mathematical issues is Szafraniec’s hallmark. His scientiﬁc contribution covers differential equations at ﬁrst, then followed by a sudden turn into functional analysis and operator theory, and then many related topics including moment problems, orthogonal polynomials, quantum physics, operators in Krein spaces and linear relations. His publication output includes around 140 papers most of which appeared in reputable journals. Szafraniec’s activity in the mathematical world together with his ability to co- operate bears fruits in many co-authored publications which encompasses numerous Mathematical work of Franciszek Hugon Szafraniec and its impacts 1299 papers written together with younger generation of mathematicians, and, as of late, physicists of all ages. Professor Szafraniec among members of Functional Analysis Group Until now Szafraniec has been an active participant of many conferences organised all over the world. His talks often go beyond a mere presentation of results, being in fact small performances interspersed with witty insertions. No wonder they gain much attention and their author is often invited to deliver speeches on many occasions, which also form the opportunity for the professor to promote Krako ´ w as a historic city which happens to thrive as a vivid scientiﬁc centre. Not only does Szafraniec attend conferences abroad, he also has been organising mathematical meetings in Krako ´ w and in other cities (like Leiden in 2009 and 2013). The conference Functions and Operators (Krako ´ w 2010) was organised on his 70th birthday while the conference 90 years of the reproducing kernel property (Krako ´ w 2000) was a celebration of his 60th birthday. It is worth noting that the reproducing kernel property in question was formulated by Stanisław Zaremba [72] who can be called Szafraniec’s scientiﬁc grandfather, as an advisor of Wazew _ ski. This motivated Szafraniec to write a unique book devoted to reproducing kernel Hilbert spaces [65]. We refer the reader to [8] for earlier biography of Szafraniec. 2 The quest for unbounded subnormality We are all normal, or at least we believe so. And we know pretty well our own limits, that is we are bounded. Linear operators acting on a Hilbert space are not so fortunate: they can deviate from normality quite a bit, being seminormal, subnormal, hyponormal, paranormal, quasinormal, not to mention that they can be unbounded. 1300 R. E. Curto et al. The dream of all spectral analysts is to deal with an operator displaying a rich spectral decomposition behavior, as close to normality as possible. Even for ﬁnite matrices this is too much to ask, not to speak about linear transformations of an inﬁnite dimensional Hilbert space. Identities such as T ¼ T ; or TT ¼ T T ¼ I; or TT ¼ T T assure the optimal spectral decomposition behavior a bounded linear operator can possess. However, these ideal and simple to verify algebraic equations are not in general met by multipliers on Hilbert spaces of analytic functions or by linear differential operators. The ﬁrst class is populated by subnormal operators, that is restrictions of normal operators to an invariant subspace, while the second one is asking for relaxing the boundedness condition, typically imposing a dense domain and a closed graph. Challenges and pitfalls abound in both situations. Ample studies covering half a century led to a good understanding of the nature of a bounded subnormal operator [12]. Even more can be said about von Neumann’s groundbreaking spectral theory of unbounded self-adjoint operators, later general- ized to other classes of linear transformations, see for instance [3]. It is the merit of Szafraniec and Stochel to have pursued with obstinacy the study of unbounded subnormal operators. They removed with high skill and in style many stumbling blocks arising on the path, even from the very beginning, starting with the deﬁnition of this class of operators. Their works excel in rigor and clarity of exposition and they are rooted in current problems of Mathematical Analysis or Quantum Physics. Their articles on this very topics are quite ramiﬁed. Without aiming at completeness we offer in the next couple of pages a glimpse into a single aspect of the Szafraniec-Stochel theory of unbounded subnormal operators. The reference list below gives a better picture of the magnitude of the strive. Let T be a linear operator with dense domain DðTÞ of a Hilbert space H. Then the adjoint operator T is deﬁned. The operator T is called formally normal if DðTÞDðT Þ and kTuk¼kT uk for all vectors u 2DðTÞ and T is said to be normal if T is formally normal and DðTÞ¼ DðT Þ. In the literature, the following characterization is often taken as a deﬁnition of normality: A dense deﬁned operator is normal if and only if T is closed and TT ¼ T T . The distinction between formally normal operators and normal operators is only important for unbounded operators, because each formally normal operator deﬁned on the whole Hilbert space is obviously bounded and normal. Normal operators stand aside. First of all, the spectral theorem holds, that is, for each normal operator T there exists a unique spectral measure E on the Borel r- algebra of C such that T ¼ zE ðzÞ. This spectral measure allows one to develop a functional calculus. For each Borel function f on the complex place there exists a unique normal operator f(T)on C deﬁned by fðTÞ¼ fðzÞdE ðzÞ. For normal operators there is a natural notion of permutability: we say two normal operators T and T acting on the same Hilbert space strongly commute if 1 2 there spectral measures commute, that is, E ðMÞE ðNÞ¼ E ðNÞE ðMÞ for T T T T 1 2 2 1 arbitrary Borel sets M, N. Further, for each n-tuple T ¼ðT ; ...; T Þ of pairwise 1 n strongly commuting normal operators there exists a unique spectral measure T on C such that T ¼ z dE ðzÞ, k ¼ 1; ...; n. This observation has immediate k k T applications to the multidimensional complex moment problem. Mathematical work of Franciszek Hugon Szafraniec and its impacts 1301 It is obvious from the deﬁnition that self-adjoint operators are normal and densely deﬁned symmetric operators are formally normal. A classical result due to Naimark [37] states that each symmetric operator has an extension to a self-adjoint operator in a possibly larger Hilbert space. In contrast, formally normal operators do not extend in general to normal operators in larger Hilbert spaces. This was ﬁrst shown by Coddington [10], see [42] for a very simple example. A densely deﬁned linear operator T on a Hilbert space H is called subnormal if there exists a normal operator N on a Hilbert space G which contains H as a subspace such that T N. Subnormal operators are formally normal. It is difﬁcult to decide whether or not a formally normal operator is subnormal or normal. The 1 d pﬃﬃ creation operator A ¼ ðx Þ of quantum mechanics is subnormal; this fact dx was nicely elaborated by Szafraniec [67] to a number of operator-theoretic characterizations of the creation operator. A systematic study of unbounded subnormal operators was begun by Stochel and Szafraniec in the mid eighties in the trilogy of fundamental papers [46–48] and continued since then in a number of research papers, see the reference list below and [68] for a leisure discussion. The ﬁrst main problem about unbounded subnormal operators is to decide whether a formally normal operator is subnormal or even normal. This is a difﬁcult problem that has many facets. There is also a natural multivariate version of the problem: It asks when a family, or a commutative -semigroup, of pairwise commuting formally normal operators has an extension to pairwise strongly commuting normal operators in a possibly larger Hilbert space. One approach is based on the presence of sufﬁciently many common ‘‘well-behaved’’ vectors. In the most general setting these are quasianalytic vectors, an idea that goes back to the work of Nussbaum [38]. In the ﬁrst paper [45] of the trilogy this was elaborated in detail and a number of basic results were obtained. Another type of characteriza- tions of subnormality based on positivity conditions is developed in [47]. All these results have natural applications to the (multidimensional) complex moment problem [50] which was both a driving force and a source of important examples for the theory. A second main problem concerns the relation of subnormal operators to its normal extensions. In particular, existence and properties of minimal extensions are important. In contrast to the bounded case, there are in general different notions of minimality. A normal extension N on a Hilbert space G of a subnormal operator T on a Hilbert space H is called minimal of spectral type if G is the only reducing subspace of N which contains H. The third work [48] of the trilogy is devoted to this area. Two kinds of minimal extensions, those of spectral type and those of cyclic type, are investigated. 3 Moment problems Inverse problems naturally occur in many branches of science and mathematics. An inverse problem entails ﬁnding the values of one or more parameters using the values obtained from observed data. A typical example of an inverse problem is the 1302 R. E. Curto et al. inversion of the Radon transform. Here a function (for example of two variables) is deduced from its integrals along all possible lines. This problem is intimately connected with image reconstruction for X-ray computerized tomography. Moment problems are a special class of inverse problems, and they arise naturally in statistics, spectral analysis, geophysics, image recognition, and economics. While the classical theory of moments dates back to the beginning of the 20th century, the systematic study of truncated moment problems began only a few years ago. In his 1987 seminal paper [32], Landau wrote, ‘‘The moment problem is a classical question in analysis, remarkable not only for its own elegance, but also for the extraordinary range of subjects, theoretical and applied, which it has illuminated’’. Szafraniec has made numerous contributions to the theory of complex moment problems, including the outstanding research reported in [49, 50, 57], and more recently the work presented in [9, 70, 71]. Some of these results include a novel approach to the complex moment problem, that is, linking positive linear functionals L acting on polynomials p in z and z with d-tuples N ðN ; ; N Þ of 1 d multiplication operators on the Hilbert space L ðlÞ, where l is a Radon measure on C . This is done using the functional calculus for normal d-tuples of operators, via the cyclic vector 1 2 L ðlÞ, as follows: LðpÞ :¼hi pðN; N Þ1; 1 ¼ pðz; zÞ dlðzÞ: This approach leads to a fruitful interplay between multivariable operator theory, the theory of positive linear functionals on the space of polynomials, and the theory of complex moment problems. Together with [44], it represents a predecessor of the unprecedented connections, beginning in the early 1990’s, among real algebraic geometry, optimization theory, the theory of quadratures in numerical analysis, the theory of moments (full and truncated), the mathematics of ﬁnance, and the theory of realizability of point processes. As a simple example of the results obtained by Szafraniec and his collaborators, we recall that the solubility of the moment problem in two variables cannot be characterized in terms of the positivity of the associated moment sequence. This is a consequence of the existence of nonnegative polynomials in two variables that do not admit a representation as a sum of squares of polynomials. In [49], the authors describe a series of additional conditions which allow a positive deﬁnite sequence to become a moment sequence, with a representing measure. These conditions have to do with the support of the representing measure, which must belong to a suitable class of algebraic curves. Along the way, the authors prove a boundedness criterion for formally normal operators in Hilbert spaces. In this way, results about moment sequences can be derived from criteria for essential normality for unbounded Hilbert space operators. The work makes contact with a pioneering result of Schmu ¨ dgen [43], which created a new bridge between operator theory and real algebraic geometry. In another trailblazing research accomplishment, in joint work with Stochel, Szafraniec discovered a polar decomposition approach to the moment problem. Mathematical work of Franciszek Hugon Szafraniec and its impacts 1303 Consider a double-indexed sequence c fc g of complex numbers, where the m;n indices m and n run over the integer lattice points of the nonnegative quarter plane; that is, whenever m; n 0. Solving the moment problem entails, in this case, ﬁnding m n a positive Borel measure l on C such that c ¼ z z dlðzÞðall m; n 0Þ.Itis m;n well known that the existence of a representing measure for c implies that c is positive deﬁnite, that is, the associated moment matrix must be positive semi- deﬁnite. It is also known that this condition is not sufﬁcient for the solubility of the complex moment problem. Now suppose that we ask c to admit a positive deﬁnite extension C to the integer lattice points of the northeast half-plane determined by the diagonal m þ n ¼ 0; that is, C must be positive deﬁnite for all pairs (m, n) such that m þ n 0. In [50], the authors proved that c has a representing measure if and only if the above mentioned extension C exists. This superb result was highlighted in a Featured Mathematical Review, alongside another superb result obtained by Putinar and Vasilescu [40]. The two articles represented outstanding additions to our existing knowledge, in terms of providing new criteria for existence and uniqueness of representing measures, and for localization of the support of such measures. They introduced original ideas, methods and techniques that had a lasting impact on subsequent developments of the theory. Both articles appealed to the notion of extendability, in different but compatible directions, and consonant with the main approach to truncated moment problems that was being developed at the time. The key ingredient needed was the idea of building a new moment problem, essentially equivalent to (and extending) the original one, but in a higher-dimensional setting, where positivity alone provides the necessary and sufﬁcient condition, just as in the single-variable case. The work in [50] was followed by a paper on determinacy and extendability [9]. More recently, Szafraniec has made substantial contributions to the study of the complex moment problem of Dirichlet type [71], and to the Sobolev moment problem [70]. Szafraniec’s ideas are often brilliant, and address fundamental problems; solutions provided indicate a profound understanding of the intrinsic structure of the mathematical entities under consideration, and of their interconnections with other areas of research. He is the type of mathematician that can make tangible and lasting connections with other sciences, esp. physics, because he truly comprehend the science and can thus create coherent and robust mathematical models to explain it. Over the years we have all enjoyed our interactions with Szafraniec at many international conferences. He is regarded as an individual with utmost wisdom, extremely sharp in his observations, and with a natural ability to focus on what is really important about a scientiﬁc matter; in particular, he has a profound understanding of the role of mathematics in science. Szafraniec has established himself as a true expert in topics ranging from moment problems to orthogonal polynomials to unbounded subnormal operator theory to dilation theory, to the theory of Krein spaces, interpolation theory, the quantum harmonic oscillator, canonical commutation relations, and so on. The high level of mathematical excellence so characteristic of Szafraniec’s early work has remained present 1304 R. E. Curto et al. throughout his long academic career, representing a clear commitment to quality of research, and making Szafraniec a model for new generations of mathematicians. 4 Reproducing kernel Hilbert spaces and positive definite kernels Functions spaces in which point evaluations are continuous with respect to their norm have been drawing the attention of mathematicians for decades. This feature is enjoyed by the so-called reproducing kernel Hilbert spaces (RKHS) which, together with the positive deﬁnite kernels they entail, play an essential role in mathematics and physics. The RKHS theory rooted in the seminal ideas of Zaremba [72] formulated at the beginning of the XX-th century and pushed forward in Aronszajn’s work [2] later on. Suppose that S is a nonempty set, and ðH; h; iÞ is a Hilbert space of complex- valued functions f on S such that the (Dirac) evaluation map d : H ! C deﬁned by d ðfÞ¼ fðxÞ is continuous for all x 2 S. For every x 2 S, the Riesz representation theorem applied to d ensures that there is a unique element K 2 H such that the x x reproducing kernel property fðxÞ¼hf ; K i holds for all f 2 H. In particular, K ðyÞ¼hK ; K i for all x; y 2 S. Then the kernel K : S S ! H deﬁned by x x y Kðx; yÞ¼ hK ; K i is a positive deﬁnite kernel in the sense that Kðx ; x Þa a > y x i j i j i;j¼1 0 for all n > 1, x ; ...; x 2 S, and a ; .. .; a 2 C. The map K is called the 1 n 1 n reproducing kernel for H. The pair ðH; KÞ is called a RKHS. The Moore- Aronszajn theorem says that there is a one-to-one correspondence between positive deﬁnite kernels and RKHS’s. Szafraniec wrote the monograph [65] in Polish on the general theory of reproducing kernel Hilbert spaces. Among important contributions of Szafraniec to the theory of RKHS and positive deﬁnite kernels we may distinguish the following. Let now S be a -semigroup with unit e, H be a Hilbert space, / : S ! BðHÞ be a positive deﬁnite map (i.e. h/ðs s Þx ; x i > 0 for all n > 1, s ; ...; s 2 S, i i j 1 n i;j¼1 j x ; ...; x 2 H) and K : S S ! BðHÞ be deﬁned by K ðt; sÞ¼ /ðs tÞ. The 1 n / / celebrated Sz.-Nagy general dilation theorem says that if the boundedness condition n n X X hK ðus ; us Þx ; x i 6 cðuÞ hK ðs ; s Þx ; x i; / i j i j / i j i j ð1Þ i;j¼1 i;j¼1 holds for all ﬁnite sequence s ; ...; s 2 S and x ; ...; x 2 H with c(u) independent 0 n 0 n of the s and x , then / can be represented in the form /ðsÞ¼ V UðsÞV , where V is a i i bounded linear map of H into a Hilbert space K, and U is a unital -preserving semigroup homomorphism from S into BðKÞ. It was Szafraniec who proved that the Sz.-Nagy general dilation theorem is equivalent to the famous Stinespring dilation theorem for completely positive operator valued linear maps on C -algebras [58]. The following inequality, sometimes called Szafraniec’s inequality states that if / is positive deﬁnite, then for all u 2 S and k > 1 (with quantiﬁers as in (1)), Mathematical work of Franciszek Hugon Szafraniec and its impacts 1305 n n 1 X X k hK ðus ; us Þx ; x i 6 hK ðs ; s Þx ; x i / i j i j / i j i j i;j¼1 i;j¼1 X k k1 k1 2 2 hK ððu uÞ s ; ðu uÞ s Þx ; x i : / i j i j i;j¼1 This inequality enabled Szafraniec to give in [55] a simple proof of the unpublished result due to Arveson saying that if / is bounded and positive deﬁnite, then / satisﬁes the boundedness condition (1). This idea culminated in proving the fol- lowing astonishing equivalent version of the boundedness condition (see [33, 56]): hK ðus; usÞx; xi 6 cðuÞhK ðs; sÞx; xi; u; s 2 S; x 2 H; ð2Þ / / and led Szafraniec to an extension of the Sz.-Nagy general dilation theorem to the case of unbounded operators by using form approach [61]. An RKHS approach to some holomorphic interpolation problems enabled him to propose a several variable analogue of the classical Pick-Nevanlinna theorem [59], which extends a result of Beatrous and Burbea [6]. This, in turn, was generalized by Quiggin and Barbian [4, 41]. In [64] he applied RKHS and its multiplication operators to model unbounded operators acting on a Hilbert space as in [48] and to investigate n-tuples of densely deﬁned operators. Hilbert C -modules are generalization of Hilbert spaces where the inner products take their values in a C -algebra instead of the ﬁeld of complex numbers. The theory of positive deﬁnite kernels in the setting of Hilbert C -modules was initiated by Murphy [36]. Szafraniec investigated Murphy’s result in [69], where he also studied the relation between the notions of complete positivity and positive deﬁniteness for C -algebras; see also [28, 35]. A series of papers [51–54] from the early 70’s develops new ideas in the representation theory of function algebras. The earlier results of Mlak and Sarason on decompositions and extensions of uniform algebras were dealing with (mostly contractive) representations in the algebra of bounded Hilbert space operators. The decompositions were obtained with respect to Gleason parts of a uniform algebra. Szafraniec has found a method based on the so called ‘‘property R’’ introduced in his papers [51] and [54] for projections belonging to the dual of Banach space. A special case of this abstract property appears in F. and M. Riesz theorem and in its abstract version on the decomposition of orthogonal measures with respect to Gleason parts of uniform algebras (see [15, II.7]). This allowed to obtain far reaching generalizations, including representations of certain non-commutative algebras on reﬂexive Banach spaces X.In[52] a spectral measure model was constructed for such representations. Another important result was a decomposition for noncontractive representations in a Hilbert space constructed with respect to a family of commuting projections having ‘‘property R’’. For such a decomposition its similarity to the orthogonal decomposition was proved in [54]. 1306 R. E. Curto et al. 5 Decompositions and extensions for operators and relations Some twenty years ago the topic of normal extensions of symmetric operators came up. Since this problem is only meaningful in the case where the symmetric operator is not densely deﬁned, it was clear that the context of the problem should be in terms of linear relations in Hilbert spaces. This lead to a number of papers written in collaboration with Hassi, Sebestye ´n and de Snoo. The original joint line of research in [23] concerned the decomposition of a linear operator from a Hilbert space H to a Hilbert space K into a regular and a singular part, as done by Jørgensen [30] and Ota [39]. Their ideas formed the right context for the decomposition of any linear relation: if H is a linear relation from H to K, i.e. a linear subspace of the product Hilbert space H K, then the adjoint relation H from K to H is deﬁned by H ¼ JH ¼ðJHÞ , where J stands for the ﬂip-ﬂop Jff ; gg¼fg; fg and the orthogonal complement is taken in the appropriate product space. The closure of H is the relation H with the multivalued part mul H (the linear space of all g 2 K such that f0; gg2 H ). Let P be the orthogonal projection from K to mul H . Then the Lebesgue decomposition of H is given by H ¼ðI PÞH þ PH, where ðI PÞH ¼fff ; ðI PÞgg : ff ; gg2 Hg; PH ¼fff ; Pgg : ff ; gg2 Hg are the regular part (a closable operator) and the singular part (its closure is a product of closed subspaces). It shares many properties with the corresponding notions from measure theory. The paper [23] was written jointly with Sebestye ´n, who would continue this line of research with his coworkers in many papers to come. More results in terms of decompositions of linear relations can be found in an issue of Dissertationes Mathematicae [24]. A relation H is called decomposable if it allows the componentwise sum decomposition H ¼ H þðf0g mul HÞg, where H is a closable operator with ran H ? mul H and where the componentwise sum 0 0 is direct. If such an operator exists, then it is automatically equal to the usual orthogonal operator part H . There is a considerable interplay between the various op operator parts and operator-like sum and componentwise sum decompositions. Recall that a linear relation H is called symmetric if H H and self-adjoint if H ¼ H . The following notions weaken these deﬁnitions: a linear relation H is called formally domain-tight if dom H dom H and domain-tight if dom H ¼ dom H . The impact of these notions on various componentwise decompositions is studied in detail. In particular, formally domain-tight and domain-tight relations are shown to play an important role in the description of Cartesian decompositions of relations (in a real and an imaginary part). Extension theory of symmetric operators or relations is to be found in [26]. If S is a bounded symmetric operator in a Hilbert space H, then the self-adjoint extensions S S H of S can be parameterized as solutions of the completion problem , where the matrix is relative to the orthogonal decomposition H ¼ dom S mul S . By choosing one bounded self-adjoint extension and deﬁning a corresponding Mathematical work of Franciszek Hugon Szafraniec and its impacts 1307 boundary triplet for S (see for instance [7]), one may characterize all self-adjoint extensions of S, including the unbounded ones. In the case where S is unbounded, there is analogous procedure when S is a maximally nondensely deﬁned operator, deﬁned by the property that kerðS kÞ\ dom S ¼f0g for all k 2 C n R; this condition is equivalent to the requirement that the symmetric relation S deﬁned by S ¼ S þðf0g mul S Þ be essentially self-adjoint, i.e., ðS Þ be self-adjoint. 1 1 Note that S is self-adjoint if and only if dom S ¼ dom S \ dom S . Under this condition all self-adjoint extensions of S are in a one-to-one correspondence to the self-adjoint relations H in the Hilbert space mul S via the perturbation formula e e H þ GHG P; where H is a ﬁxed self-adjoint extension of S, which is transversal with S , P is the orthogonal projection from H onto mul S , and G is a boundedly invertible operator from H onto mul S . Observe that S is the Friedrichs extension of S when S is semibounded. The topic of extension theory in Hilbert spaces is revisited in [27]. Let S be a symmetric relation in a Hilbert space H. An extension H of S is said to be intermediate if S H S . The self-adjoint extensions of a symmetric relation are by deﬁnition all automatically intermediate; they have been described by Coddington [11]. What about normal extensions? First note that a linear relation H in H is called normal if there exists an isometry V from H onto H of the form Vff ; gg¼ff ; hg, where ff ; gg2 H and ff ; hg2 H . It is known that H is normal if and only if H is closed and H H ¼ HH .If S is densely deﬁned, then the normal extensions of S are automatically self-adjoint and they are all operators. If S is nondensely deﬁned there may be normal extensions H of S which are not intermediate. Necessary and sufﬁcient conditions for the existence of normal nonself-adjoint intermediate extensions are developed and all such extensions are parameterized. The perturbation formula from [26] plays an important role. The work [23] with Szafraniec has inﬂuenced further work along these lines: it sufﬁces to mention [22], where the original Lebesgue decompositions have been extended to a more general context. Furthermore, the joint work [34] has had a direct inﬂuence on [21]. Columns, rows, and blocks are now introduced in [21] not only for operators but also for relations; a simple example of this was already encountered in [26]. Furthermore, Szafraniec was one of the editors of [25] and he turned out to be extremely conscientious and precise in that capacity. 6 Coherent states Professor Szafraniec’s interest in coherent states stems from his research concerning basic problems of the operator theory and functional analysis. More than 20 years ago this led him to studies of non-canonical operator structures used to describe generalized models of quantum harmonic oscillator [60, 66], as well as to an extension of his scientiﬁc activity to investigation of reproducing kernel Hilbert spaces (RKHS) emerging in quantum physics, in particular coming there from considerations involving coherent states. Mathematical background of the coherent states theory were efforts oriented to problems underlying quantization and unaware using methods of RKHS which motivated Klauder [31] and Bargmann [5]to 1308 R. E. Curto et al. introduce in the early 60’ of the previous century harmonic oscillator coherent states (HOCS) as new tools of quantum physics initially treated mere as mathematical objects. Soon after discovered applicability of HOCS in quantum optics gained them high popularity which resulted in more and more extensive investigations of HOCS and their generalizations going beyond those relevant to the standard harmonic oscillator. Two features of coherent states, both standard and generalized, specially attracted Szafraniec’s attention. The ﬁrst was their utility for constructing the Segal- Bargmann transform as an unitary map which is generated by coherent states and links the Hilbert space of square integrable Schrodinger wave functions L ðR; dxÞ with the Hilbert space L ðC; lðdzÞÞ \ H of entire functions while the second was hol assigning mathematically correct meaning to the so-called resolution of the identity interpreted by the physicists community as sufﬁcient condition enabling them to use coherent states as overcomplete non-orthogonal reference frames. Szafraniec soon recognized that the right tool to analyze and understand both problems is to apply the RKHS methods and become one of the forerunners of such methodology [62, 65]. The novel feature of Szafraniec’s approach was to use his author’s formalism of the RKHS theory. The crucial elements which enable to apply this formalism are: i.) to introduce a set of complex valued functions fU ðxÞg of x 2 X which satisfy U ðxÞU ðxÞ\1 for all x (if it happens, U ðxÞ’s are called the kernel functions) n n n and ii) to construct the reproducing kernel according to Kðx; yÞ¼ U ðxÞU ðyÞ. n n Having the set fU ðxÞg one may proceed further and construct prospective generalized coherent states c ¼ U ðxÞe with the Hilbert space basis fe g x n n n n n entering the game. In the Section ‘‘Komentarze’’ (‘‘Comments’’) closing the Chapter 3 of his book [65] entitled Reproducing kernel Hilbert spaces (unfortu- nately available only in the Polish edition published by the Jagiellonian University) Szafraniec emphasized the validity of the RKHS structure for the construction of the Segal-Bargmann space and wrote‘‘It should be noted here that kernel functions in the Segal-Bargmann space are an example of coherent states, the notion appearing in the quantum mechanics’’. The RKHS approach to the theory of generalized coherent states, being during last ten years developed by Szafraniec in collaboration with Ali, Go ´ rska and Horzela, provided interesting results. Here should be mentioned a new look at the resolution of the identity understood in the context of the Segal-Bargmann transform [29] and explicit construction of the single particle and bipartite Hermite coherent states built with holomorphic Hermite polynomials of single and two variables used to construct suitable kernel functions [13, 20]. Such formed coherent states obey physically interesting properties—for single variable case they are the squeezed states of quantum optics [1] while for two variable case represent quantum states which minimize the Heisenberg uncertainty relations, i.e. are squeezed-coherent, and (what is astonishing) at the same time entangled, i.e. non-factorizable, which signalizes coexistence of coherence and non-dynamical correlations [19]. Another problem of the coherent states theory studied by Szafraniec (together with Gorska and Horzela) was inconsistency of naive generalization of squeezed yk states disturbing physicists for many years [14]. The operator S ðnÞ¼ expðna k Mathematical work of Franciszek Hugon Szafraniec and its impacts 1309 na Þ used to deﬁne HOCS for k ¼ 1 and the squeezed states for k ¼ 2 becomes meaningless if k 3 as its matrix elements are given by divergent series. This happens although the operator in the exponent remains antihermitean which many physicists are used to treat as implying the unitarity of S. Szafraniec and his yk k collaborators explained this puzzle showing that the operator na na has deﬁciency indices equal to 0 as long as k ¼ 1; 2 and thus it is self-adjoint. But for k [ 2 its deﬁciency indices are equal to (k, k) so it is only essentially self-adjoint and S ðnÞ is not unitary [18]. If lacks unitarity it does not ﬁt to the standard quantum mechanical formalism and should not be used in any routine way. 7 Quantization Theoretical physicists do not give the same constraints as would do a mathematician when writing an article. This means that physicist’s approach to problems is often more intuitive than mathematically demonstrative. One could ﬁnd in Szafraniec a very open-minded and helpful character, able to seize fully physicist’s intuition while fully respecting mathematical rigor. One could learn a lot from Szafraniec when considering problems pertaining to the formalism of Quantum Physics. Szafraniec fully understands subtleties of quantum formalism and is able to share this understanding with a large community of physicists. On the other hand, he is a pure mathematician, with outstanding expertise in unbounded operator theory, moment problems, orthogonal polynomials, and in rigorous formulation of the essence of Quantum Mechanics, namely the canonical commutation rule. To be more precise here, the major mathematical contribution of Szafraniec on the latter question is a characterization, in terms of subnormality, of the canonical solution (creation-annihilation) of the commutation relation of the quantum harmonic oscillator. Since the advent of quantum mechanics one recurrent question concerns the transition from classical to quantum models (i.e., quantization) for some system, regardless its physical relevance. One interesting method is to use generalized coherent states and more generally positive operator valued measure (POVM) to implement what is named integral quantization. To some extent, the latter is similar to the Berezin–Toeplitz quantization. In a nutshell, the Berezin–Toeplitz quanti- zation of a symplectic manifold M with Ka ¨hler structure maps functions on M to operators in the Hilbert space of square-integrable holomorphic sections of an appropriate complex line bundle. Denoting by P the orthogonal projection operator from the space of all square-integrable sections to the holomorphic subspace, for any bounded measurable function f, one constructs the Toeplitz operator A with symbol f, acting on the space of holomorphic sections, as A / ¼ Pðf/Þ. That is, A / consists of multiplication by f followed by projection back into the holomorphic subspace. Two Szafraniec’s papers, [62, 63] were at the origin of two articles devoted to this integral quantization with coherent states. In the ﬁrst one, [16], a coherent state quantization of the complex plane was presented when the latter is equipped with a non rotationally invariant measure. While the canonical commutation rule (up to a simple rescaling) still holds true, these authors explained 1310 R. E. Curto et al. how the involved coherent states, built from holomorphic continuations of Hermite polynomials, are related to the non-commutative plane. In the second paper, [17], they examined mathematical questions around angle (or phase) operator associated with a number operator through a short list of basic requirements, and they have implemented three methods of construction of quantum angle. The ﬁrst one is based on operator theory and parallels the deﬁnition of angle for the upper half-circle through its cosine and completed by a sign inversion. The two other methods are based on the integral quantization with adapted coherent states. Now, a basic requirement in the construction of coherent states is the resolution of identity, which usually invokes an appropriate measure. In the process of generalization of coherent states, it may be advantageous to have a construction which does not explicitly make use of a measure. As a matter of fact, a measure-free construction was developed in other Szafraniec’s papers, like [29]. The key point is the existence of a sequence of complex functions satisfying a certain convergence criterion. The reproducing kernel Hilbert space, required for the coherent states, can be constructed out of these functions. Examples are provided where these sequences appear, e.g. in moment problems and orthogonal polynomials. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/. References 1. Ali, S.T., Go ´ rska, K., Horzela, A., Szafraniec, F.H.: Squeezed states and Hermite polynomials in a complex variable. J. Math. Phys. 55(1), 012107 (2014) 2. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950) 3. Bade, W.G.: Unbounded spectral operators. Pac. J. Math. 4, 373–392 (1954) 4. Barbian, C.: A characterization of multiplication operators on reproducing kernel Hilbert spaces. J. Oper. Theory 65, 235–240 (2011) 5. Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure Appl. Math. 14, 187–214 (1961) 6. Beatrous, F., Burbea, J.: Positive-deﬁniteness and its applications to interpolation problems for holomorphic functions. Trans. Am. Math. Soc. 284, 247–270 (1984) 7. Behrndt, J., Hassi, S., de Snoo, H.S.V.: Boundary Value Problems, Weyl Functions, and Differential Operators. Monographs in Mathematics vol. 108, Birkha ¨user (2020) 8. Cichon ´ , D., Littlejohn, L., Stochel, J.: Franciszek Hugon Szafraniec: a scholar of eminence. Complex Anal. Oper. Theory 6, 529–531 (2012) 9. Cichon, D., Stochel, J., Szafraniec, F.: The complex moment problem: determinacy and extendibility. Math. Scand. 124, 263–288 (2019) 10. Coddington, E.A.: Formal normal operators having no normal extension. Can. J. Math. 17, 1030–1040 (1965) 11. Coddington, E.A.: Extension Theory of Formally Normal and Symmetric Subspaces. Mem. Am. Math. Soc. 134 (1973) Mathematical work of Franciszek Hugon Szafraniec and its impacts 1311 12. Conway, J.B.: The Theory of Subnormal Operators. Mathematical Surveys and Monographs, 36. American Mathematical Society, Providence, RI (1991) 13. van Eijndhoven, S.J.L., Meyers, J.L.H.: New orthogonality relations for the Hermite polynomials and related Hilbert spaces. J. Math. Anal. Appl. 146, 89–98 (1990) 14. Fisher, R.A., Nieto, M.M., Sandberg, V.D.: Impossibility of naively generalizing squeezed coherent states. Phys. Rev. D 29, 1107–1110 (1984) 15. Gamelin, T.W.: Uniform Algebras. Prentice Hall Inc, Englewood Clifs, NJ (1969) 16. Gazeau, J.-P., Szafraniec, F.H.: Holomorphic Hermite polynomials and a non-commutative plane. J. Phys. A 44, 495201 (2011). 13 pp 17. Gazeau, J.-P., Szafraniec, F.H.: Three paths toward the quantum angle operator. Ann. Phys. 375, 16–35 (2016) 18. Gorska, K., Horzela, A., Szafraniec, F.H.: Squeezing: the ups and downs. Proc. R. Soc. A 470, 20140205 (2014) 19. Gorska, K., Horzela, A., Szafraniec, F.H.: Coherence, squeezing and entanglement - an example of peaceful coexistence, Ch. 5, pp. 89–117, in Coherent states and their applications: A contemporary panorama (Springer, Berlin, 2018) 20. Go ´ rska, K., Horzela, A., Szafraniec, F.H.: Holomorphic Hermite polynomials in two variables. J. Math. Anal. Appl. 470, 750–772 (2019) 21. Hassi, S., Labrousse, J.-Ph., de Snoo, H.S.V.: Operational calculus for rows, columns, and blocks of linear relations. Adv. Oper. Theory (to appear) 22. Hassi, S., Sebestyen, Z., de Snoo, H.S.V.: Lebesgue type decompositions for linear relations and Ando’s uniqueness criterion. Acta Sci. Math. (Szeged) 84, 465–507 (2018) 23. Hassi, S., Sebestyen, Z., de Snoo, H.S.V., Szafraniec, F.H.: A canonical decomposition for linear operators and linear relations. Acta Math. Hungar. 115, 281–307 (2007) 24. Hassi, S., de Snoo, H.S.V., Szafraniec, F.H.: Componentwise and canonical decompositions of linear relations. Dissertationes Mathematicae 465 (2009) 25. Hassi, S., de Snoo, H.S.V., Szafraniec, F.H. (editors): Operator methods for boundary value prob- lems. Lecture Notes 404, London Mathematical Society (2012) 26. Hassi, S., de Snoo, H.S.V., Szafraniec, F.H.: Inﬁnite-dimensional perturbations, maximally non- densely deﬁned symmetric operators, and some matrix representations. Indag. Math. 23, 1087–1117 (2012) 27. Hassi, S., de Snoo, H.S.V., Szafraniec, F.H.: Normal intermediate extensions of symmetric relations. Acta Math. Szeged 80, 195–232 (2014) 28. Heo, J.: Reproducing kernel Hilbert C -modules and kernels associated with cocycles. J. Math. Phys. 49, 103507 (2008). 12 pp 29. Horzela, A., Szafraniec, F.H.: A measure-free approach to coherent states. J. Phys. A 45, 244018 (2012). 9 pp 30. Jørgensen, P.E.T.: Unbounded operators: perturbations and commutativity problems. J. Funct. Anal. 39, 281–307 (1980) 31. Klauder, J.R.: The action option and a Feynman quantization of spinor ﬁelds in terms of ordinary c-numbers. Ann. Phys. 11, 123–168 (1960) 32. Landau, H.J.: Classical background of the moment problem, in Moments in Mathematics. Proceeding of Symposia Application Mathematics, 37, AMS Short Course Lecture Notes, American Mathe- matics Society, Providence, RI, pp. 1–15 (1987) 33. Masani, P.: Dilations as propagators of hilbertian varieties. SIAM J. Math. Anal. 9, 414–456 (1978) 34. Mo ¨ ller, M., Szafraniec, F.H.: Adjoints and formal adjoints of matrices of unbounded operators. Proc. Am. Math. Soc. 136, 2165–2176 (2008) 35. Moslehian, M.S.: Conditionally positive deﬁnite kernels in Hilbert C -modules. Positivity 21, 1161–1172 (2017) 36. Murphy, G.J.: Positive deﬁnite kernels and Hilbert C -modules. Proc. Edinburgh Math. Soc. (2) 40, 367–374 (1997) 37. Naimark, M.A.: Spectral functions of a symmetric operator. Izv. Akad. Nauk 7, 285–296 (1943) 38. Nussbaum, A.E.: Quasi-analytic vectors. Ark. Mat. 6, 179–191 (1971) 39. Ota, S.: On a singular part of an unbounded operator. Z. Anal. Anwendungen 7, 15–18 (1988) 40. Putinar, M., Vasilescu, F.-H.: Solving moment problems by dimensional extension. Ann. Math. 149, 1087–1107 (1999) 41. Quiggin, P.: For which reproducing kernel Hilbert spaces is Pick’s theorem true? Integr. Equ. Oper. Theory 16, 244–266 (1993) 1312 R. E. Curto et al. 42. Schmu ¨ dgen, K.: A formally normal operator without normal extension. Proc. Am. Math. Soc. 98, 503–504 (1985) 43. Schmu ¨ dgen, K.: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289, 203–206 (1991) 44. Stochel, J.: Moment functions on real algebraic sets. Ark. Mat. 30, 133–148 (1992) 45. Stochel, J., Szafraniec, F.H.: A characterization of subnormal operators, Spectral theory of linear operators and related topics (Timi soara/Herculane, 1983), 261–263, Oper. Theory Adv. Appl., 14, Birkha ¨user, Basel (1984) 46. Stochel, J., Szafraniec, F.H.: On normal extensions of unbounded operators. I. J. Oper. Theory 14, 31–55 (1984) 47. Stochel, J., Szafraniec, F.H.: On normal extensions of unbounded operators. II. Acta Sci. Math. (Szeged) 53, 153–177 (1989) 48. Stochel, J., Szafraniec, F.H.: On normal extensions of unbounded operators. III. Publ. RIMS Kyoto Univ. J. 25, 105–139 (1989) 49. Stochel, J., Szafraniec, F.: Algebraic operators and moments on algebraic sets. Portugal. Math 51, 25–45 (1994) 50. Stochel, J., Szafraniec, F.H.: The complex moment problem and subnormality: a polar decomposition approach. J. Funct. Anal. 159, 432–491 (1998) 51. Szafraniec, F.H.: Decomposition of operator valued representations of Banach algebras. Bull. Acad. Polon. Sci. Se ´r. Sci. Math. Astron. Phys. 18, 321–324 (1970) 52. Szafraniec, F.H.: Some spectral properties of operator-valued representations of function algebras. Ann. Polon. Math. 25, 187–194 (1971) 53. Szafraniec, F.H.: Orthogonal decompositions of non-contractive operator valued representations of Banach algebras. Bull. Acad. Polon. Sci., Se ´r. Sci. Math. Astron. Phys. 19, 937–940 (1971) 54. Szafraniec, F.H.: Decompositions of non-contractive operator-valued representations of Banach algebras. Stud. Math. 42, 97–108 (1972) 55. Szafraniec, F.H.: On the boundedness condition involved in a general dilation theory. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 24, 877–881 (1976) 56. Szafraniec, F.H.: A propos of professor Masani’s talk. In: Probability theory on vector spaces, Proceedings, Trzebieszowice, Poland 1977 (Ed.: A. Woron). Lecture Notes in Mathematics 656, Springer, Berlin, pp. 245–249 (1978) 57. Szafraniec, F.H.: Boundedness of the shift operator related to positive deﬁnite forms: an application to moment problems. Ark. Mat. 19, 251–259 (1981) 58. Szafraniec, F.H.: Dilations of linear and nonlinear maps, in Functions, Series, Operators, Proceed- ings, Budapest (Hungary), 1980, Sz.-Nagy, B., Szabados, J. (eds.), Colloquia Mathematica Societatis Janos Bolyai, vol. 35, pp. 1165–1169, North-Holland, Amsterdam (1983) 59. Szafraniec, F.H.: On bounded holomorphic interpolation in several variables. Monatsh. Math. 101, 59–66 (1986) 60. Szafraniec, F.H.: A RKHS of entire functions and its multiplication operator. An explicit example, in Linear Operators in Function Spaces, Proceedings, Timis ¸oara (Romania), June 6-16, 1988. Helson, H., Sz. Nagy, B., Vasilescu, F.-H. (eds.) Operator Theory: Advances and Applications, vol. 43, pp. 309–312, Birkha ¨user, Basel (1990) 61. Szafraniec, F.H.: The Sz.-Nagy ‘‘the ´ore `me principal’’ extended. Application to subnormality. Acta Sci. Math. (Szeged) 57, 249–262 (1993) 62. Szafraniec, F.H.: Analytic models of the quantum harmonic oscillator. Contemp. Math. 212, 269–276 (1998) 63. Szafraniec, F.H.: Subnormality in the quantum harmonic oscillator. Commun. Math. Phys. 210, 323–334 (2000) 64. Szafraniec, F.H.: The reproducing kernel Hilbert space and its multiplication operators. In: Complex analysis and related topics (Cuernavaca, 1996), 253–263, Oper. Theory Adv. Appl., 114, Birkha ¨user, Basel (2000) 65. Szafraniec, F.H.: Przestrzenie Hilberta z ja ˛drem reprodukuja ˛cym (Reproducing kernel Hilbert ´ ´ spaces), in Polish, Wydawnictwo Uniwersytetu Jagiellonskiego, Krakow (2004) 66. Szafraniec, F.H.: Operators of the q-oscillator. Banach Center Publ. 78, 293–307 (2007) 67. Szafraniec, F.H.: How to recognize the creation operator. Rep. Math. Phys. 59, 401–408 (2007) 68. Szafraniec, F.H.: Normals, subnormals and an open question. Oper. Matrix. 4, 485–510 (2010) Mathematical work of Franciszek Hugon Szafraniec and its impacts 1313 69. Szafraniec, F.H.: Murphy’s Positive deﬁnite kernels and Hilbert C -modules reorganized. In: Non- commutative harmonic analysis with applications to probability II, 275–295, Banach Center Publ. 89, Polish Acad. Sci. Inst. Math., Warsaw (2010) 70. Szafraniec, F., Wojtylak, M.: The Sobolev moment problem and Jordan dilations. J. Math. Anal. Appl. 444, 1675–1689 (2016) 71. Szafraniec, F., Wojtylak, M.: The complex moment problem of Dirichlet type, submitted (2020) 72. Zaremba, S.: L’e ´quation biharmonique et une classe remarquable de fonctions fondamentales har- moniques. Bulletin International de l’Academie des Sciences de Cracovie 147–196 (1907) Afﬁliations 1 2 3 • • • Rau´ l E. Curto Jean-Pierre Gazeau Andrzej Horzela 4 5,6 7 • • • Mohammad Sal Moslehian Mihai Putinar Konrad Schmu¨ dgen 8 9 Henk de Snoo Jan Stochel Rau ´ l E. Curto raul-curto@uiowa.edu Jean-Pierre Gazeau gazeau@apc.in2p3.fr Andrzej Horzela Andrzej.Horzela@ifj.edu.pl Mohammad Sal Moslehian moslehian@um.ac.ir; moslehian@yahoo.com Mihai Putinar mputinar@math.ucsb.edu; mihai.putinar@ncl.ac.uk Konrad Schmu ¨ dgen schmuedgen@math.uni-leipzig.de Henk de Snoo hsvdesnoo@gmail.com The University of Iowa, Iowa City, IA 52242, USA APC (UMR 7164), Department of Physics, Universite Paris-Diderot, 75205 Paris, France Department of Mathematical Physics, H. Niewodniczan ´ ski Institute of Nuclear Physics, Polish Academy of Sciences, ul. Eljasza-Radzikowskiego 152, 31342 Krako ´ w, Poland Department of Pure Mathematics, Ferdowsi University of Mashhad, Center of Excellence in Analysis on Algebraic Structures (CEAAS), P.O. Box 1159, Mashhad 91775, Iran University of California at Santa Barbara, Santa Barbara, CA, USA Newcastle University, Newcastle Upon Tyne, UK Mathematical Institute, University of Leipzig, Augustusplatz 10, 04109 Leipzig, Germany Bernoulii Institute for Mathematics, Computer Science and Artiﬁcial Intelligence, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands Instytut Matematyki, Uniwersytet Jagiellon ´ ski, ul. Łojasiewicza 6, PL 30348 Krako ´ w, Poland

Advances in Operator Theory – Springer Journals

**Published: ** Jul 8, 2020

Loading...

You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!

Read and print from thousands of top scholarly journals.

System error. Please try again!

Already have an account? Log in

Bookmark this article. You can see your Bookmarks on your DeepDyve Library.

To save an article, **log in** first, or **sign up** for a DeepDyve account if you don’t already have one.

Copy and paste the desired citation format or use the link below to download a file formatted for EndNote

All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.