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D. Voiculescu (1985)
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We study joint numerical and spectral radii defined for d-tuples of bounded oper- ators on a Hilbert space and related to noncommutative notions of independence. The definitions are in analogy with the ones of Popescu, where his formulations turned out to be related with free creation operators, and in this way related to the free independence of Voiculescu. In our study the definitions are related with either weakly monotone creation operators, and thus associated with the monotone inde- pendence of Muraki, or with boolean creation operators, and hence related with the boolean independence. Keywords Numerical radius Spectral radius Fock space Monotone independence Boolean independence Mathematics Subject Classification 46L53 47A12 47A13 1 Introduction The notion of numerical radius as well as the related notion of numerical range is an object of intensive studies since the work by Toeplitz [10] in 1918 until today. Numerical radius provides a norm on the space of bounded operators, which is equivalent to the operator norm. Its special features include unitarity invariance, the Communicated by Lyudmila Turowska. Dedicated to Franciszek Hugon Szafraniec on the occasion of his 80th birthday, with appreciation of his influence on our multidimensional collaboration. & Janusz Wysoczanski janusz.wysoczanski@math.uni.wroc.pl Anna Kula anna.kula@math.uni.wroc.pl Institute of Mathematics, University of Wrocław, Wrocław, Poland 1040 A. Kula and J. Wysoczan´ski power inequality and the relation with the spectral radius, see e.g. [6]. The numerical range and radius link the properties of operators with geometry of a complex plane, allowing many interesting applications, from the approximate localization of spectrum via the stability results for differential equations (e.g. [5]) to the von Neumann type inequalities (e.g. [1]). In 2009, Gelu Popescu [9], in his search for a free analogue of the Sz.-Nagy- Foias theory for row contractions, defined free analogues of numerical and spectral radii, namely the joint numerical radius and joint spectral radius for d-tuples of operators ðT ; ...; T Þ acting on a Hilbert space H. The definitions are as follows. 1 d Definition 1 The (free) joint numerical radius is 8 9 < d = X X X w ðT ; ...; T Þ :¼ sup hT h jh i : kh k \1 F 1 d j g a a a : ; þ þ j¼1 a2F a2F d d þ þ where F is the free semigroup with free generators g ; ...; g and each a 2 F is a 1 d d d word in these generators. Definition 2 The (free) joint spectral radius is 1=2k r ðT ; ...; T Þ :¼ lim T T F 1 d a k!1 jaj¼k where for a 2 F one puts jaj¼ k if it is a word in k generators a ¼ g ...g . i i d 1 k These two notions are related not only to the free semigroup F , but also to the model of freeness of Voiculescu [11], and more precisely to the creation operators on the full Fock space by the following result. Theorem 3 ([9], Corollary 1.2) The joint free numerical and spectral radii can be computed as the ordinary numerical radius w and the spectral radius r of single operators: w ðT ; ...; T Þ¼wðS T þ ... þ S TÞð1Þ F 1 d 1 d 1 d r ðT ; ...; T Þ¼rðS T þ ... þ S T Þ; ð2Þ F 1 d 1 d 1 d where S :¼ Sðe Þ; ...; S :¼ Sðe Þ are the free creation operators on the full Fock 1 1 d d space FðH Þ on d-dimensional Hilbert space H with an orthonormal basis d d fe ; ...; e g and S(h) is the creation operator by the vector h 2 H : SðhÞv ¼ h v 1 d d for v 2FðH Þ. The main idea of this paper is to study analogues of these definitions in the case where we replace the (free) full Fock space by a Fock space associated to other noncommutative independences, and the free creation operators by the creation operators on the appropriate Fock space. We show that the joint (noncommutative) numerical radii, defined in analogy to Theorem 1, satisfy many basic properties similar to the free case. In particular, we show the unitarity invariance and the relation with the Joint monotone and boolean numerical and spectral radii 1041 appropriately defined spectral radius. We also compute some examples. In the paper we treat the monotone and boolean case, but the framework is more general. Our idea establishes yet another bridge between classical operator theory and the noncommu- tative probability and we believe this is a starting point for further investigations. However, in this paper there is no need to define the monotone and boolean independences, it is sufficient to consider the models of both of them, built on either weakly monotone Fock space or on the Boolean Fock space, respectively. 2 General scheme Recall that the (classical) numerical radius of a linear operator T, bounded on a Hilbert space H, is defined by w ðTÞ :¼ supfjhTh; hij : h 2H; khk¼ 1g cl and the (classical) spectral radius of T is defined by r ðTÞ¼ lim kT k : cl m!1 Let us consider one of the noncommutative independence, e.g. boolean, free, monotone and let us consider a noncommutative Fock space F ðHÞ associated to this independence. By this we mean the full Fock space ([11]) for the free inde- pendence of Voiculescu, the weakly monotone Fock space ([12]) for the monotone independence of Muraki and the boolean Fock space ([3]) for the boolean inde- pendence (c.f. [2]). On each of these Fock spaces we are given creation and anni- hilation operators, which we shall use to define relative joint (numerical and spectral) radii, following the work by Popescu [9] for the free case. Let d 2 N and let H be the d-dimensional Hilbert space with the orthonormal H H basis fe ; 1 j dg. Denote by A ; ...; A the creation operators associated to the 1 d H H basic vectors A ¼ A ðe Þ on the related Fock space F ðHÞ and define the H-joint j H numerical radius of the d-tuple ðT ; ...; T Þ of operators in BðHÞ by 1 d H H w ðT ; .. .; T Þ :¼ w ðA T þ ...A T Þ: H 1 d cl 1 1 d d Similarly, we define the H-joint spectral radius of ðT ; ...; T Þ by 1 d H H r ðT ; .. .; T Þ :¼ r ðA T þ .. .A T Þ: H 1 d cl 1 1 d d The following properties of the H-joint numerical and spectral radii are immediate consequences of the construction and of the properties of the classical numerical radius (compare with Theorem 1.1 in [9]). Proposition 4 The H-joint numerical radius and joint spectral radius satisfy: (i) w ðkT ; ...; kT Þ¼ jkjw ðT ; ...; T Þ for any k 2 C; H 1 d H 1 d 0 0 0 0 (ii) w ðT þ T ; ...; T þ T Þ w ðT ; ...; T Þþ w ðT ; ...; T Þ; H 1 d H 1 d H 1 d 1 d (iii) w ðU T U; ...; U T UÞ¼ w ðT ; ...; T Þ for any unitary U : K!H; H 1 d H 1 d 1042 A. Kula and J. Wysoczan´ski (iv) w ðI T ; ...; I T Þ¼ w ðT ; .. .; T Þ for any separable Hilbert space H K 1 K d H 1 d K; (v) r ðT ; ...; T Þ w ðT ; ...; T Þ. H 1 d H 1 d Proof The properties (i) and (ii) follows from w ðkTÞ¼jkjw ðTÞ and cl cl 0 0 w ðT þ T Þ w ðTÞþ w ðT Þ. As for (iii), for a unitary U : K!H we set V ¼ cl cl cl H H I U and A :¼ A T þ ...A T and observe that 1 1 d d H H w ðU T U; ...; U T UÞ¼ w ðA U T U þ ...A U T UÞ H 1 d cl 1 1 d d H H ¼ w ðI UÞ ðA T þ ...A T ÞðI UÞ ¼ w ðV AVÞ cl cl 1 1 d d ¼ supfjhAVh; Vhij : h 2H; khk¼ 1g 0 0 0 0 ¼ supfjhAh ; h ij : h ¼ Vh 2F ðHÞK; kh k¼kVhk¼ 1g ¼ w ðAÞ¼ w ðT ; ...; T Þ: cl H 1 d The property (iv) goes exactly as in the proof of [9, Theorem 1.1], using w ðI TÞ¼ w ðTÞ: cl K cl w ðI T ; ...; I T Þ¼ w A ðI T Þ H K 1 K d cl K j j¼1 ! ! d d X X H H ¼ w I ð A T Þ ¼ w A T cl K j cl j j j j¼1 j¼1 ¼ w ðT ; ...; T Þ: H 1 d Finally, to show (v) we just use the classical result r ðTÞ w ðTÞ. h cl cl 3 Joint boolean numerical and spectral radii Recall after [3] that the boolean Fock space (over the d-dimensional space H)is defined as the direct sum F ðHÞ¼ CX H; ð3Þ where X is a unit vector, called the vacuum. The boolean creation and annihilation operators are given by f if h ¼ X 0if h ¼ X B ðfÞh ¼ ; BðfÞh ¼ : ð4Þ 0if h 2 H hf ; hiX if h 2 H For a fixed orthonormal basis fe : 1 j dg in H, we shall use the notation B :¼ B ðe Þ, B :¼ Bðe Þ, j ¼ 1; .. .; d, for the creation and annihilation operators j j j (respectively) by the basic vectors, and e :¼ X. It is easy to see that B B ¼ d P , 0 j jk X where P is the projection onto the vacuum vector e ¼ X. X 0 Let now ðT ; ...; T Þ be the d-tuple of bounded operators on a Hilbert space H. 1 d We define the joint boolean numerical radius of ðT ; ...; T Þ as 1 d Joint monotone and boolean numerical and spectral radii 1043 w ðT ; ...; T Þ :¼ w ðB T þ ...B T Þ B 1 d cl 1 1 d d and the spectral joint boolean spectral radius of ðT ; ...; T Þ as 1 d r ðT ; ...; T Þ :¼ r ðB T þ ...B T Þ: B 1 d cl 1 1 d d We first provide the explicit formula for computing the joint boolean numerical radius, which is the analogue of Popescu’s definition in the free case (see Definition 1). Proposition 5 The joint boolean numerical radius can be expressed as () d d X X w ðT ; ...; T Þ¼ sup hg ; T g i : g ; g ; ...; g 2H; kg k ¼ 1 : B 1 d 0 j j 0 1 d j j¼1 j¼0 Proof By the definition of the classical numerical radius, we have w ðT ; ...; T Þ :¼ supfjh B T h; hij : h 2F ðHÞH; khk¼ 1g: B 1 d B j j j¼1 Expressing h 2F ðHÞH as h ¼ e g with g ; g ; ...; g 2H (recall B k k 0 1 d k¼0 that e :¼ X), we observe that d d d d X X X X 2 2 khk ¼h e g ; e g i¼ he ; e ihg ; g i¼ kg k ; k k m m k m k m k k¼0 m¼0 k;m¼0 k¼0 whereas, using the relation B e ¼ d e , for 1 j; m d, and B e ¼ 0 for j m jm 0 j 0 1 j d, we get d d d X X X h B T h; hi¼ hðB T Þðe g Þ; ðe g Þi k k m m j j j j j¼1 j¼1 k;m¼0 d d X X ¼ hB e ; e ihT g ; g i k m k m j j j¼1 k;m¼0 d d X X ¼ he ; B e ihT g ; g i k j m k m j¼1 k;m¼0 d d XX ¼ he ; e ihg ; T g i k 0 k m m m¼1 k¼0 d d X X ¼ hg ; T g i¼hg ; T g i: 0 m m 0 m m m¼1 m¼1 h 1044 A. Kula and J. Wysoczan´ski It turns out that all of the properties that were shown to hold for the (free) joint numerical radius (see [9, Theorem 1.1]), remains true in the boolean case. Some of them were already observed in Proposition 4; here we prove the remaining ones. Proposition 6 We have ðiÞ w ðT ; ...; T Þ¼ 0 if and only if T ¼ ... ¼ T ¼ 0, B 1 d 1 d P 1 P 1 d d ðiiÞ 1 2 2 B k T T k w ðT ; ...; T Þk T T k , j B 1 d j j¼1 j j¼1 j ðiiiÞ w ðX T X; ...; X T XÞkXk w ðT ; ...; T Þ for any bounded operator B B 1 d B 1 d X : H!K. Proof Ad ðiÞ . By the properties of the classical numerical radius, w ðT ; ...; T Þ¼ B 1 d P P d d 0 implies B T ¼ 0. Hence for any h ¼ e g 2F ðHÞH, one k k B j¼1 j j k¼0 gets d d d d d X XX X X 0 ¼ B T h ¼ B e T g ¼ B e T g ¼ e T g : k k 0 0 j 0 j j j j j j j j¼1 j¼1 k¼0 j¼1 j¼1 Thus, for any j ¼ 1; ...; d and g 2H, one gets T g ¼ 0. Consequently, T ¼ 0 0 0 j j and, equivalently, T ¼ 0 for any j ¼ 1; ...; d. Ad ðiiÞ . Thanks to kTk w ðTÞkTk, we have cl d d X X w ðT ; ...; T Þ¼ w ð B T Þ B T B 1 d cl j j j j j¼1 j¼1 d d d X X X ¼ð B T Þ ð B T Þ ¼ B B T T k k j j k k j j |ffl{zffl} j¼1 k¼1 j;k¼1 ¼d P jk X 1 1 2 2 d d X X ¼ P T T ¼kP k T T : X j X j j j |fflffl{zfflffl} j¼1 j¼1 ¼1 Similarly, we prove that w ðT ; ...; T Þ T T . B 1 d j j¼1 j Ad ðiiiÞ . Let X : H!K be a bounded operator and let g ; ...; g 2H satisfy 0 d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P d 2 d 2 kg k ¼ 1. Define C :¼ kXg k and h :¼ Xg , k ¼ 0; 1; ...; d. k k k k k¼0 k¼0 C d 2 Then kh k ¼ 1 and C kXk. Consequently, k¼0 Joint monotone and boolean numerical and spectral radii 1045 () d d X X w ðX T X; ...; X T XÞ¼ sup hXg ; T Xg i : kg k ¼ 1 B 1 d 0 j j j j¼1 j¼0 () d d X X sup hCh ; T Ch i : kh k ¼ 1 0 j j j j¼1 j¼0 2 2 ¼ C w ðT ; ...; T ÞkXk w ðT ; ...; T Þ: B 1 d B 1 d Remark 7 The properties (i), (ii) and ðiÞ show that the joint boolean numerical radius is a norm on BðHÞ , which, by ðiiÞ , is actually equivalent to the operator norm T T of the operator row matrix ½T ; ...; T , hence w is a continuous j 1 d B j¼1 map in the norm topology. We now compute some examples and, in particular, show that w 6¼ w . B F Example 8 For T ¼ I , j ¼ 1; ...; d we have j H pffiffiffi w ðI ; ...; I Þ¼ : B 1 d Using the relation between the ‘ - and ‘ -norm on H : 1 2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d d X uX pffiffiffi kg k d kg k for g ; ...; g 2H; k k 1 d k¼1 k¼1 and Proposition 5, we observe that vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d d d d X X X uX pffiffiffi jhg ; g ij kg kk g kkg k kg k dkg k kg k : 0 k 0 k 0 k 0 k k¼1 k¼1 k¼1 k¼1 d 2 Hence for any g ; ...; g 2H such that kg k ¼ 1, denoting 0 d k k¼0 t :¼kg k2 ½0; 1, we get vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d d pffiffiffi uX X 2 2 w ðI ; ...; I Þ supf dkg k kg k : g ; ...; g 2H; kg k ¼ 1g B 1 d 0 k 0 d k k¼1 k¼0 pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ¼ supft d 1 t : t 2½0; 1g ¼ ; pffiffi 1 d pffiffi and the supremum is achieved when t ¼ . This show that w ðI ; ...; I Þ . B 1 d 2 g 1 0 pffiffi To see that the equality holds, take g 2H with kg k ¼ and g ¼ , 0 0 k pffiffi P P d 2 d d d pffiffi k ¼ 1; ...; d. Then kg k ¼ 1 whereas jhg ; g ij ¼ jhg ; g ij ¼ . k 0 k 0 0 k¼1 k¼1 2 d 1046 A. Kula and J. Wysoczan´ski Example 9 For T ¼ B , j ¼ 1; ...; d, the boolean annihilators, we have j j pffiffiffi w ðB ; ...; B Þ¼ : B 1 d Since B h ¼hh; e iX for h 2H, we compute j j d d d X X X jhg ; B g ij ¼ jhg ; hg ; e iXij kg kj hg ; e ij 0 j j 0 j j 0 j j j¼1 j¼1 j¼1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u pffiffiffi d d X X pffiffiffi u kg k kg k dkg k kg k 0 j 0 j j¼1 j¼1 as shown in the previous Example. d 2 1 1 pffiffi pffiffiffiffi Taking g ¼ X and g ¼ e for j ¼ 1; ...; d, we find that kg k ¼ 1 0 k j k k¼1 2 2d while pffiffiffi d d X X 1 d jhg ; B g ij ¼ pffiffiffi jhX; he ; e iXi¼j : 0 j j j j 2 d j¼1 j¼1 Example 10 For d ¼ 1 we find out that 1 1 ð5Þ w ðTÞ¼ w ðTÞ w ðTÞ w ðTÞ¼ w ðTÞ F cl B cl F 2 2 and w ðTÞ¼ w ðTÞ when H¼ C, Tz :¼ az, a 2 C . This shows that B F w ðT ; ...; T Þ 6¼ w ðT ; ...; T Þ in general. F 1 d B 1 d It was shown in [9, Sect. 1] that w ðTÞ¼ w ðTÞ. Now, observe that F cl no 2 2 w ðTÞ¼ sup hg ; Tg i : g ; g 2H; kg k þkg k ¼ 1 B 0 1 0 1 0 1 sup hg ; Tg i : g 2H; kg k ¼ 0 0 0 0 pffiffiffi 1 1 1 ¼ sup hh; Thi : h :¼ 2g 2H; khk ¼ ¼ w ðTÞ: 0 cl 2 2 2 2 2 On the other hand, given g ; g 2H, with kg k þkg k ¼ 1, we can repeat the 0 1 0 1 ih idea of Popescu, setting f :¼ g þ e g for h 2½0; 2p. Then h 0 1 Joint monotone and boolean numerical and spectral radii 1047 Z Z 2p 2p 2 ih ih kf k dh ¼ hg þ e g ; g þ e g idh h 0 1 0 1 0 0 Z Z Z 2p 2p 2p 2 2 2 ih ih ¼kg k dh þhg ; g ik e dh þhg ; g ik e dh 0 0 1 1 0 0 0 0 2p þkg k dh 2 2 ¼ 2pðkg k þkg k Þ¼ 2p; 0 1 and Z Z 2p 2p ih ih ih ih hf ; Tf ie dh ¼ hg þ e g ; Tðg þ e g Þie dh h h 0 1 0 1 0 0 Z Z Z Z 2p 2p 2p 2p ih 2ih ih ¼ hg ; Tg ie dh þ hg ; Tg idh þ hg ; Tg ie dh þ hg ; Tg ie dh 0 0 0 1 1 0 1 1 0 0 0 0 ¼ 2phg ; Tg i: 0 1 So, using the fact that jhh; Thij w ðTÞkhk , we get cl Z Z 2p 2p 1 1 ih hg ; Tg i jhf ; Tf ie jdh w ðTÞ kf k dh ¼ w ðTÞ: 0 1 h h cl h cl 2p 2p 0 0 For the special choice H¼ C and T z :¼ az for some fixed a 2 C , we get w ðT Þ¼ w ðT Þ. Indeed, B a cl a no i/ iw z ¼ e sin t; w ¼ e cos t; 2 2 w ðT Þ¼ sup hz; awi : kzk þkwk ¼ 1 ¼ B a /; w; t 2½0; 2p jaj jaj w ðT Þ cl a ¼ supfj sinð2tÞj; t 2½0; 2pg ¼ ¼ : 2 2 2 It is an open problem to check if the equality w ðTÞ¼ w ðTÞ can hold. B F We end this Section with an observation that the joint boolean spectral radius degenerates. Proposition 11 The joint boolean spectral radius is always 0. Proof Since the boolean creation operators satisfy B B ¼ 0 for any j, k, we have j k r ðT ; ...; T Þ¼ rðB T þ ...B T Þ¼ lim kð B T k B 1 d 1 1 d d j j m!1 j¼1 ¼ lim k B ...B T ...T k ¼ 0: j j j j 1 m 1 m m!1 j ;...;j ¼1 1 m h 1048 A. Kula and J. Wysoczan´ski 4 Joint monotone numerical and spectral radii We consider the model of the monotone independence on the discrete weakly monotone Fock space F ðHÞ, defined in [13]. This space is built upon a d- WM dimensional Hilbert space H with a given orthonormal basis fe : 1 j dg,asa closed subspace of the full Fock space FðHÞ, spanned by the vacuum vector X :¼ e and the simple tensors e ... e , where the indices are in weakly 0 j j k 1 monotonic order: 1 j ... j . By the standard convention we identify e h ¼ 1 k 0 h e ¼ h for any h 2F ðHÞ. 0 WM The creation operator M by the vector e is defined as follows: M ðe ... e Þ¼ e e ... e if j j ... j 1; j j j j j k 1 j k 1 k 1 M ðe ... e Þ¼0if j\j : j j k j k 1 The annihilation operator is defined as M ðe Þ¼ X; M ðe ... e Þ¼ d e ... e ; j j j j j jj j j k 1 k k 1 1 and they are mutually adjoint: ðM Þ ¼ M . It is useful to introduce the following orthogonal projections. By definition P is the orthogonal projection onto e ¼ X and for m 1: P ðe ... e Þ¼ e ... e if m ¼ j ... j 1; m j j j j k 1 k 1 k 1 P ðe ... e Þ¼0if m 6¼ j and m 1; m j j k k 1 Q :¼ P þ P þ ... þ P : m 0 1 m Then Q is the orthogonal projection onto the span of e ¼ X and vectors of the m 0 form e ... e with j m, and P is the orthogonal projection onto the span of j j k m k 1 vectors of the form e ... e with j ¼ m, k 1. j j k k 1 The weakly monotone creation and annihilation operators satisfy the following commutation relations M M ¼ M M ¼0if j\k ð6Þ k j j k M M ¼0if j 6¼ k ð7Þ M M ¼ Q for 1 k d ð8Þ k k M M ¼ P for 1 k d: ð9Þ k k In particular, for j\k we have M M M M ¼ M Q M ¼ Q : ð10Þ j k j k j k j j Remark 12 The monotone creation operators are bounded and generate - subalgebras which are monotone independent in the sense of Muraki [7, 8] (for the proof see [13] and [4]). Joint monotone and boolean numerical and spectral radii 1049 In this setting the joint monotone numerical radius is defined as w ðT ; ...; T Þ :¼ w ðM T þ ...M T Þ M 1 d cl 1 1 d d and the joint monotone spectral radius of ðT ; ...; T Þ is defined as 1 d r ðT ; ...; T Þ :¼ r ðM T þ ...M T Þ: M 1 d cl 1 1 d d To provide explicit formulas for the joint monotone numerical and spectral radii we introduce some operations on weakly monotone sequences. For each k 2 N :¼ N [f0g we define M :¼fði ; ...; i Þ2 N : d i ... i 1g; k 1 k k 1 k 1 M :¼f0g M :¼f;g k þ d 1 Note that each M with k 1 is finite with cardinality . We also set M :¼ M to be the union of all M for k 0 and M :¼M[ M . k k 1 k 0 There is a natural comparison relation R of the sequences from M, namely for a ¼ði ; ...; i Þ2 M and b :¼ðj ; ...; j Þ2M with k; m 1we put k 1 k m 1 m ða; bÞ2 R if i j 1 m ða; bÞ 62 R if i \j 1 m ða; 0Þ2 R if a 2M ; k 1: Observe that this relation is neither symmetric nor antisymmetric. For convenience we shall write a b iff ða; bÞ2 R. Definition 13 The weakly monotone concatenation of such sequences is defined on M as follows: Let a ¼ði ; ...; i Þ2M and b :¼ðj ; ...; j Þ2M with k; m 1, k 1 k m 1 m then ab :¼ði ; ...; i ; j ; ...; j Þ if a b k 1 m 1 ab :¼; if ða; bÞ 62 R; 0a ¼ a0 :¼a ;a ¼ a; :¼;: If a 2M , b 2M with k; m 0 and a b, then ab 2M . k m kþm In particular, for j 2M and a ¼ði ; ...; i Þ2M we shall have ja ¼ 1 k 1 k ðj; i ; ...; i Þ if j a. As one can see for this concatenation 0 plays the role of k 1 neutral element and the empty set ; behaves like 0 in multiplication of numbers. Then, with the notation 1050 A. Kula and J. Wysoczan´ski 0if a ¼;; e ¼ X if a 2M ; a 0 e ... e if a ¼ði ; ...; i Þ2M ; k 1; i i k 1 k k 1 the set fe : a 2Mg is an orthonormal basis of F ðHÞ. We shall extend this a WM notation to the operators: if T ; ...; T 2 BðHÞ are given and a 2M, then 1 d I if a 2M ; T ¼ T ... T if a ¼ði ; .. .; i Þ2M ; k 1: i i k 1 k k 1 Now we are ready to describe the explicit formula for the joint monotone numerical radius of d-tuple of operators. Proposition 14 Let ðT ; ...; T Þ be a d-tuple of bounded operators on a Hilbert 1 d space H. Their joint monotone numerical radius can be computed as () X X X w ðT ; ...; T Þ¼ sup hg jT g i : g 2H; kg k ¼ 1 : ð11Þ M 1 d a j ja a a j¼1 a2M a2M Proof We express h 2F ðHÞH as h ¼ e g with g 2H. Then M a a a a2M w ðT ; .. .; T Þ¼ supfjh M T h; hij : h 2F ðHÞH; khk¼ 1g M 1 d M j j j¼1 () X X X X ¼ sup jh M T ð e g Þ; ð e g Þij : g 2 h; k e g k¼ 1 : a a b b a a a j j j¼1 a2M b2M a2M Since X X X k e g k ¼h e g ; e g i a a a a b b a2M a2M b2M X X ¼ he ; e ihg ; g i¼ kg k ; a b a b a a;b2M a2M and *+ X X X M T ð e g Þ; ð e g Þ ¼ a a b b j j j¼1 a2M b2M X X ¼ hM e ; e ihT g ; g i a b a b j j j¼1 a;b2M d d X X X X ¼ he ; e ihT g ; g i¼ hg ; T g i; ja b a b a j ja j¼1 a;b2M j¼1 a2M the formula (11) follows. h Joint monotone and boolean numerical and spectral radii 1051 Note that in the summation over a 2M the nonzero terms might be only for those a for which ja 6¼;, i.e. when j i if a ¼ði ; ...; i Þ. For example, if j ¼ 1 k k 1 then it must be a ¼ 0or a ¼ð1; ...; 1Þ, so in this case the summation over a 2M reduces to the summation over k ¼ 0; 1; ..., the length of the sequence of 1’s. To study the properties of the joint monotone numerical radius we need the following lemma. Lemma 1 For any T ; ...; T 2 BðHÞ we have 1 d d d X X ð12Þ M T ¼ max T T : j j j 1 m d j¼1 j¼m Proof Using (7) and (8) and the definition of Q , we observe that d d d d X X X X M T ¼ð M T Þð M T Þ ¼ M M T T j j j j j j k k k k |fflffl{zfflffl} j¼1 j¼1 k¼1 j;k¼1 ¼d Q jk j 1 1 2 j 2 d d X X X ¼ Q T T ¼ P T T j j m j j j j¼1 j¼1 m¼0 d d d X X X ¼ P T T þ P T T 0 j m j j j j¼1 m¼1 j¼m Now, we use the mutual orthogonality of the orthogonal projections P ; P ; ...; P 0 1 d to get d d X X M T ¼ max T T : j j j 1 m d j¼1 j¼m In analogy to Proposition 6 we have the following properties of the joint monotone numerical radius. Proposition 15 We have ðiÞ w ðT ; ...; T Þ¼ 0 if and only if T ¼ ... ¼ T ¼ 0, M 1 d 1 d d d ðiiÞ X X M 1 1 1 2 2 max k T T k w ðT ; ...; T Þ max k T T k , j M 1 d j j j 1 m d 1 m d j¼m j¼m ðiiiÞ w ðX T X; ...; X T XÞkXk w ðT ; ...; T Þ for any bounded operator M 1 d M 1 d X : H!K. 1052 A. Kula and J. Wysoczan´ski Proof Ad ðiÞ . As in the proof of Proposition 6, w ðT ; .. .; T Þ¼ 0 implies M 1 d P P d d M T ¼ 0. Hence for any h ¼ e g 2F ðHÞH, one gets k k B j¼1 j j k¼0 d d d d X XX XX 0 ¼ M T h ¼ M e T g ¼ e e T g : k k j k 0 j j j j j j¼1 j¼1 k¼0 k¼0 j [ k Hence T ¼ 0 for any j ¼ 1; ...; d. Ad ðiiÞ . Due to w ðTÞkTk and Lemma 1 we have cl d d d X X X w ðT ; ...; T Þ¼ w ð M T Þ M T ¼ max T T : M 1 d cl j j j j j j 1 m d j¼1 j¼1 j¼m The other part follows from kTk w ðTÞ. cl Ad ðiiiÞ . Let X : H!K be a bounded operator and let g ; ...; g 2H satisfy 0 d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P P d 2 d 2 kg k ¼ 1. Define C :¼ kXg k and h :¼ Xg , k ¼ 0; 1; ...; d. k k k k k¼0 k¼0 d 2 Then kh k ¼ 1 and C kXk. Consequently, k¼0 () d d X X w ðX T X; ...; X T XÞ¼ sup hXg ; T Xg i : kg k ¼ 1 M 1 d 0 j j j j¼1 j¼0 () d d X X sup hCh ; T Ch i : kh k ¼ 1 0 j j j j¼1 j¼0 2 2 ¼ C w ðT ; .. .; T ÞkXk w ðT ; .. .; T Þ: M 1 d M 1 d Remark 16 The properties ðiÞ , ðiiÞ and ðiÞ show that the joint monotone M M M numerical radius is a norm on BðHÞ , which, by ðiiÞ , is actually equivalent to the maximum, over 1 m d, of the operator norms of the operator row matrices ½T ; ...; T . m d Example 17 For d ¼ 1 and T 2 BðHÞ we get w ðTÞ¼ w ðTÞ¼ w ðTÞ. M F cl This follows immediately from the fact that for d ¼ 1 the set M consists of the unique term ð1; ...; 1Þ (k-times). Hence, () 1 1 X X w ðTÞ¼ sup hg ; Tg i : kg k ¼ 1 ; M k kþ1 k k¼0 k¼0 which is exactly the same as the joint free numerical radius for a single operator, and which in turn was shown to be equal to the classical numerical radius (see [9, Sect. 1]). The joint monotone spectral radius can be computed similarly to the (free) joint spectral radius of Popescu. Joint monotone and boolean numerical and spectral radii 1053 Proposition 18 The joint monotone spectral radius satisfies 1=2k X X r ðT ; ...; T Þ¼ lim max T T ; M 1 d a k!1 1 m d j¼m a2M where M :¼fa ¼ði ; ...; i Þ2M : i ... i ¼ jg ð13Þ k 1 k k 1 is a set of monotone sequences of length k, starting from j. Proof First observe that d d X X ð M T ¼ M ...M T ...T j j j j j j k 1 k 1 j¼1 j ;...;j ¼1 1 k X X ¼ M ...M T ...T ¼ M T ; j j j j a a k 1 k 1 j ... j 1 a2M k 1 k due to (6). This also implies d d X X k k ð M T ð M T m m j j m¼1 j¼1 d d X X ¼ M ...M M ...M T ...T T ...T m m m m 1 k j j 1 k j j k 1 k 1 m ;...;m ¼1 j ;...;j ¼1 1 k 1 k X X ¼ M ...M M ...M T ...T T ...T m m m m 1 k j j 1 k j j k 1 k 1 1 m ... m j ... j 1 1 k k 1 ¼ M ...M M ...M T ...T T .. .T j j j j 1 k j j 1 k j j k 1 k 1 j ... j 1 k 1 ¼ Q T ...T T ...T j j j 1 1 k j j k 1 j ... j 1 k 1 X X ¼ Q T T ...T T ...T T m m j j 2 k j j m k 2 m¼1 m j ... j d 2 k X X ¼ ðQ þ P þ ... þ P Þ T T ...T T ...T T 1 2 m m j j 2 k j j m k 2 m¼1 m j ... j d 2 k ¼ Q T T ...T T .. .T T þ 1 1 j j 2 k j j 1 k 2 1 j ... j d 2 k d d X X X þ P T T .. .T T ...T T m i j j 2 k j j i k 2 m¼2 i¼m i j ... j d 2 k 1054 A. Kula and J. Wysoczan´ski Since the projections Q ; P ; ...; P are mutually orthogonal, we obtain 1 2 d d d X X k k ð M T ð M T m m j j m¼1 j¼1 X X ¼ max T T .. .T T ...T T : j j j 1 2 k j j j k 2 1 1 m d j ¼m j j ... j d 1 1 2 k Therefore we can write d d X X r ðT ; ...; T Þ¼ r ð M T Þ¼ lim k M T k M 1 d cl j j j j k!1 j¼1 j¼1 1=2k d d X X k k ¼ lim ð M T ð M T m m j j k!1 m¼1 j¼1 1=2k X X ¼ lim max T T ...T T ...T T j j j 1 2 k j j j k 2 1 k!1 1 m d j ¼m j j ... j d 1 1 2 k 1=2k X X ¼ lim max T T : k!1 1 m d j¼m a2M 4.1 Example: joint monotone numerical radius for weakly monotone annihilation operators We shall compute the joint monotone numerical radius for d-tuple of weakly monotone annihilators M ; ...; M . An upper bound for w ðM ; .. .; M Þ is easily 1 d M 1 d obtained. Proposition 19 For any d 2 we have w ðM ; ...; M Þ d: M 1 d Proof Using the fact that the norm of any creation operator M is 1 and the Cauchy- Schwarz inequality, we have the following d d d X X X X X X hg jM g i jhM g jg ij kg kkg k a j ja a ja a ja j¼1 a2M j¼1 a2M j¼1 a2M d d X X X X 1 1 2 2 2 2 kg k kg k 1 ¼ d a ja j¼1 a2M a2M j¼1 Joint monotone and boolean numerical and spectral radii 1055 for any family ðg Þ satisfying kg k ¼ 1. h a a a2M a2M This is a rough estimate and the following calculations show that perhaps the pffiffiffi optimal one would be related to d. To support this claim, we provide lower estimates of that sort. It will be instructive to treat first the case d ¼ 2. Proposition 20 For two weakly monotone annihilation operators T ¼ M and 1 1 pffiffiffi pffiffiffi pffiffiffi T ¼ M , we have w ðM ; M Þ 2 0; 77 2. 2 2 M 1 2 3 3 Proof For the proof we shall analyze in details the expression XX hg jM g i: a j ja j¼1 a For this purpose we shall use the notation a ¼½k; m if a ¼ð2; ...; 2; 1; .. .1Þ is a sequence of m times the digit 2 followed by k times the digit 1. Then we have explicit formulas for the composition ja with j ¼ 1; 2, namely 2½k; m¼ ½k; m þ 1 if k; m 0, 1½k; 0¼ ½k þ 1; 0 if k 1 and 1½k; m¼ 0if m 1. We shall also denote the orthonormal basis e ¼ e if a ¼½k; m; in particular a m;k he je i¼ 1 if and only if s ¼ u; r ¼ t. Each vector g can be written in the s;r u;t ½k;m r;s orthonormal basis as g ¼ g e and then we get ½k;m s;r ½k;m r;s 0 X X 2 r;s 2 kg k ¼ jg j ½k;m ð14Þ a k;m;r;s 0 and 2 2 XX X X X X r;s t;u hg jM g i¼ g g he jM e i a j ja s;r j u;t ½k;m j½k;m j¼1 a j¼1 k;m 0 r;s 0 t;u 0 ð15Þ X X X X r;s t;u ¼ g g hM e je i: s;r u;t ½k;m j½k;m j¼1 k;m 0 r;s 0 t;u 0 We split this summation into two parts. 1. For j ¼ 1 we have necessarily m ¼ 0 and hM e je i¼he je i if s ¼ 0 s;r u;t 0;rþ1 u;t and hM e je i¼ 0if s 1. Hence for j ¼ 1 we get only the following nonzero s;r u;t term XX X X r;s t;u r;0 rþ1;0 g g he je i¼ g g : 0;rþ1 u;t ½k;0 ½kþ1;0 ½k;0 ½kþ1;0 k 0 r 0 t;u 0 k;r 0 2. For j ¼ 2 and m 0 in a similar manner we get hM e je i¼he je i¼ 1 if and only if u ¼ s þ 1; t ¼ r: s;r u;t sþ1;r u;t 2 1056 A. Kula and J. Wysoczan´ski Hence the nonzero term in (15)is X X X X X X X r;s t;u r;s r;sþ1 g g he je i¼ g g : sþ1;r u;t ½k;m ½k;mþ1 ½k;m ½k;mþ1 k 0 m 1 r 0 t;u 0 k 0 m 1 r;s 0 So we get 2 2 XX X X X X r;s t;u hg jM g i¼ g g hM e je i a j ja s;r u;t ½k;m j½k;mÞ j j¼1 j¼1 k;m 0 r;s 0 t;u 0 X X X r;0 rþ1;0 r;s r;sþ1 ¼ g g þ g g : ½k;0 ½kþ1;0 ½k;m ½k;mþ1 k;r 0 k;m 0 r;s 0 Now we consider special vectors, defined for a constant x 2ð0; 1Þ and k; m; r; s 0: mþk x ; if r ¼ k; s ¼ m rs g ¼ ð16Þ ½k;m 0 otherwise We have that in this case the quantity in (14) equals 1 1 X X X 2 rs 2 2k 2m kg k ¼ jg j ¼ x x ¼ : a ð17Þ ½k;m ð1 x Þ k;m;r;s¼0 k;m¼0 a2M On the other hand, we have XX hg jM g i a j ja j¼1 a X X X 2kþ1 mþk mþkþ1 ¼ x þ x x ð18Þ k 0 k 0 m 0 x x 2x x ¼ þ ¼ : 2 2 2 2 2 1 x ð1 x Þ ð1 x Þ Hence maximizing the quotient of (18)by(17) we get the function fðxÞ :¼ 2x x , qffiffi qffiffi 4 2 2 which in (0, 1) has the local maximum at x ¼ . 3 3 3 To sum up, we have rffiffiffi 4 2 w ðM ; M Þ sup fðxÞ¼ : M 1 2 3 3 x2ð0;1Þ We are now ready to treat the general case. Proposition 21 For d 3 and the monotone annihilators T ¼ M , j ¼ 1; ...; d, we j j have Joint monotone and boolean numerical and spectral radii 1057 pffiffiffi w ðM ; ...; M Þ d: M 1 j Proof We generalize the idea from the proof of the case d ¼ 2. For a fixed d 3, any a 2M is of the form a ¼ðd; ...; d; d 1; ...; d 1 ; ...; 2; ...; 2; 1; ...; 1Þ; |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl} k times k times k times k times d d 1 2 1 where k ; ...; k 0. In the sequel, such an element will be simply denoted by d 1 a ¼½k ; ...; k . For arbitrary (but fixed) x 2ð0; 1Þ we shall consider the vectors d 1 k þ...þk d 1 g ¼ g :¼ x e ; a ½k ;...;k a d 1 for which we have X X 2 2ðk þ...þk Þ d 1 kg k ¼ x ¼ : ð19Þ ð1 x Þ k ;...;k 0 a2M d 1 Now, we would like to compute X X hg jM g i: ð20Þ a j ja j¼1 a2M For this purpose observe that, for a fixed 1 j d,if a ¼½k ; ...; k , then ja 6¼; if d 1 and only if there are no indices bigger than j in a, that is if k ¼ 0; ...; k ¼ 0. In jþ1 d such case ja ¼½0; ...; 0; k þ 1; k ; ...; k and j j 1 1 hg jM g i¼hg jM g i a j ja j ½0;...;0;k ;...;k ½0;...;0;k þ1;...;k j 1 j 1 k þ...þk k þ...þk þ1 j 1 j 1 ¼hx e jx M e i ½0;...;0;k ;...;k j ½0;...;0;k þ1;...;k j 1 j 1 2ðk þ...þk Þþ1 2ðk þ...þk Þþ1 j 1 j 1 ¼ x he je i¼ x : 0;...;0;k ;...;k ½0;...;0;k ;...;k j 1 j 1 Otherwise ja ¼; and hg jM g i¼ 0. a j ja This simplifies the expression (20) considerably: d d d X X X X X 2ðk þ...þk Þþ1 j 1 hg jM g i¼ x ¼ : ð21Þ a j ja ð1 x Þ j¼1 a2M j¼1 k ;...k 0 j¼1 j 1 Taking into account the normalization (19) of the family ðg Þ , we arrive to the problem of maximizing the function: d d 1 j X X j¼1 2 ð1 x Þ 2 d j 2 k f ðxÞ :¼ ¼ xð1 x Þ ¼ x ð1 x Þ ð22Þ j¼1 k¼0 ð1 x Þ over (0, 1). Setting y ¼ 1 x this reduces to the maximization of the function 1058 A. Kula and J. Wysoczan´ski d 1 pffiffiffiffiffiffiffiffiffiffiffiX 1 y g ðyÞ :¼ 1 y y ¼ pffiffiffiffiffiffiffiffiffiffiffi 1 y k¼0 over (0, 1). Trying to resolve the optimization problem explicitly, one arrives to the equation d d 1 ð23Þ ð2d 1Þy 2dy þ 1 ¼ 0; which describes the critical points of g . We shall not go this way. We just observe that the value of g at the points y ¼ 1 provides the required estimates. d d Namely, pffiffiffi pffiffiffi 1 1 5 sup gðyÞ gð1 Þ¼ d 1 ð1 Þ d for d 3: d d 9 y2ð0;1Þ This shows that the numerical radius satisfies pffiffiffi w ðM ; ...; M Þ sup fðxÞ¼ sup gðyÞ d: M 1 d x2ð0;1Þ y2ð0;1Þ Let us remark that numerical calculations reveal that the point y ¼ 1 seems to be a good approximation of y , at which the maximum of g is achieved. d;max d Moreover, for 3 d 100, the relative error appearing when g ðy Þ is replaced d d;max by g ðy Þ is smaller than 2%. d d All these calculations and observations have led us to the formulation of the following statement. Conjecture 1 There exists an absolute constant c [ 0 such that the joint monotone numerical radius of the d-tuple of weakly monotone annihilators equals pffiffiffi w ðM ; ...; M Þ¼ c d: M 1 d 4.2 Example: joint monotone spectral radius for weakly monotone creation operators Using Proposition 18 we compute the joint monotone spectral radius of d weakly monotone annihilation operators M ; ...; M . In this case we have 1 d 1=2k X X r ðM ; ...; M Þ¼ lim max M M ¼ 1: M 1 d a k!1 1 m d j¼m a2M j j Indeed, let us denote by c the cardinality of the set M , see (13). Then k k k þ d j j j c ¼jM j¼ . Since for a 2M , we have M M ¼ Q , the right- jk a j k k a k 1 hand side above can be written as Joint monotone and boolean numerical and spectral radii 1059 1=2k 1=2k d d j X X X j j lim max c Q ¼ lim max c ðP þ P Þ j 0 i k k k!1 1 m d k!1 1 m d j¼m j¼m i¼1 8 9 1=2k 1=2k < d m d m d m = X X X mþj mþj ¼ lim max Q c ; P c m mþs k k k!1 1 m d: ; j¼0 s¼1 j¼s 8 9 1=2k 1=2k < d m d m = X X mþj mþj ¼ lim max Q c ; max P c m mþs k k k!1 1 m d: 1 s d m ; j¼s j¼0 8 9 ! ! 1=2k 1=2k < d m d m = X X mþj mþj ¼ lim max c ; max c k k k!1 1 m d: 1 s d m ; j¼0 j¼s We see that the maximum is achieved for m ¼ 1 and the last expression equals 8 9 ! ! 1=2k 1=2k < d m = d X X mþj j lim max max c ¼ lim c k k k!1 1 m d 0 s d m: ; k!1 j¼s j¼1 1=2k 1=2k k þ d j k þ d ¼ lim ¼ lim ¼ 1: k!1 k!1 k 1 k 1 j¼1 5 Concluding remarks and open problems This paper is a beginning of the studies of joint numerical and spectral radii related to creation operators independent in noncommutative sense: the monotone and boolean ones. We have showed that some properties are analogous as in the work of Popescu for free creation operators. However, especially for the monotone case, which we base on the weakly monotone creation operators, our results differ from the free case. There is still a lot about the noncommutative joint numerical and spectral radii to be understood. Some open problems appeared in our study, in particular to find the exact formulas for the joint monotone numerical radius of d 2 weakly monotone pffiffiffi annihilation operators (at first glance it seems to be related to d rather then to d). But to see the real power of these objects one should search for analogues of the classical power inequality or von Neumann type inequalities. Finally, it would be interesting to know what are the relations between the three types (free, boolean and monotone) of numerical radii. 1060 A. Kula and J. Wysoczan´ski Acknowledgements AK was supported by the Polish National Science Center Grant SONATA 2016/21/ D/ST1/03010. JW was supported by the Polish National Science Center Grant OPUS 2016/21/B/ST1/ 00628. AK and JW were also supported by the Polish National Agency for Academic Exchange (NAWA) within the POLONIUM program PPN/BIL/2018/1/00197/U/00021. Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest. 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. 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Advances in Operator Theory – Springer Journals
Published: Jul 2, 2020
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