Joint monotone and boolean numerical and spectral radii of d-tuples of operators

We study joint numerical and spectral radii defined for d-tuples of bounded operators 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 independence of Muraki, or with boolean creation operators, and hence related with the boolean independence.


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 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 1 ; . . .; T d Þ acting on a Hilbert space H. The definitions are as follows.
Definition 1 The (free) joint numerical radius is hT j h g j a jh a i : X where F þ d is the free semigroup with free generators g 1 ; . . .; g d and each a 2 F þ d is a word in these generators.

Definition 2 The (free) joint spectral radius is
where for a 2 F þ d one puts jaj ¼ k if it is a word in k generators a ¼ g i 1 . . .g i k . These two notions are related not only to the free semigroup F þ d , 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 F ðT 1 ; . . .; T d Þ ¼wðS 1  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 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 noncommutative 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.

General scheme
Recall that the (classical) numerical radius of a linear operator T, bounded on a Hilbert space H, is defined by w cl ðTÞ :¼ supfjhTh; hij : h 2 H; khk ¼ 1g and the (classical) spectral radius of T is defined by Let us consider one of the noncommutative independence, e.g. boolean, free, monotone and let us consider a noncommutative Fock space F H ðHÞ associated to this independence. By this we mean the full Fock space ( [11]) for the free independence of Voiculescu, the weakly monotone Fock space ( [12]) for the monotone independence of Muraki and the boolean Fock space ( [3]) for the boolean independence (c.f. [2]). On each of these Fock spaces we are given creation and annihilation operators, which we shall use to define relative joint (numerical and spectral) radii, following the work by Popescu [9] for the free case. Let Similarly, we define the H-joint spectral radius of ðT 1 ; . . .; T d Þ by 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: Proof The properties (i) and (ii) follows from w cl ðkTÞ ¼ jkjw cl ðTÞ and w cl ðT þ T 0 Þ w cl ðTÞ þ w cl ðT 0 Þ. As for (iii), for a unitary U : The property (iv) goes exactly as in the proof of [9, Theorem 1.1], using w cl ðI K TÞ ¼ w cl ðTÞ: Finally, to show (v) we just use the classical result r cl ðTÞ w cl ðTÞ. h

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 where X is a unit vector, called the vacuum. The boolean creation and annihilation operators are given by For a fixed orthonormal basis fe j : 1 j dg in H, we shall use the notation B Ã j :¼ B Ã ðe j Þ, B j :¼ Bðe j Þ, j ¼ 1; . . .; d, for the creation and annihilation operators (respectively) by the basic vectors, and e 0 :¼ X. It is easy to see that B j B Ã k ¼ d jk P X , where P X is the projection onto the vacuum vector e 0 ¼ X.
Let now ðT 1 ; . . .; T d Þ be the d-tuple of bounded operators on a Hilbert space H. We define the joint boolean numerical radius of ðT 1 ; . . .; T d Þ as and the spectral joint boolean spectral radius of ðT 1 ; . . .; T d Þ as 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 Proof By the definition of the classical numerical radius, we have whereas, using the relation B j e m ¼ d jm e 0 , for 1 j; m d, and B j e 0 ¼ 0 for h 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
Proof Ad ðiÞ B . By the properties of the classical numerical radius, w B ðT 1 ; . . .; Ad ðiiÞ B . Thanks to 1 2 kTk w cl ðTÞ kTk, we have Ad ðiiiÞ B . Let X : H ! K be a bounded operator and let g 0 ; . . .; The properties (i), (ii) and ðiÞ B show that the joint boolean numerical radius is a norm on BðHÞ d , which, by ðiiÞ B , is actually equivalent to the operator of the operator row matrix ½T 1 ; . . .; T d , hence w B is a continuous map in the norm topology.
We now compute some examples and, in particular, show that w B 6 ¼ w F . : Using the relation between the ' 1 -and ' 2 -norm on H d : and Proposition 5, we observe that Hence for any g 0 ; . . .; g d 2 H such that P d k¼0 kg k k 2 ¼ 1, denoting t :¼ kg 0 k 2 ½0; 1, we get and the supremum is achieved when t ¼ as shown in the previous Example. Taking On the other hand, given g 0 ; g 1 2 H, with kg 0 k 2 þ kg 1 k 2 ¼ 1, we can repeat the idea of Popescu, setting f h :¼ g 0 þ e ih g 1 for h 2 ½0; 2p. Then So, using the fact that jhh; Thij w cl ðTÞkhk 2 , we get hg 0 ; Tg 1 i 1 2p For the special choice H ¼ C and T a z :¼ az for some fixed a 2 C Ã , we get w B ðT a Þ ¼ 1 2 w cl ðT a Þ. Indeed, It is an open problem to check if the equality w B ðTÞ ¼ w F ðTÞ can hold. 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 Ã j B Ã k ¼ 0 for any j, k, we have

Joint monotone numerical and spectral radii
We consider the model of the monotone independence on the discrete weakly monotone Fock space F WM ðHÞ, defined in [13]. This space is built upon a ddimensional Hilbert space H with a given orthonormal basis fe j : 1 j dg, as a closed subspace of the full Fock space F ðHÞ, spanned by the vacuum vector X :¼ e 0 and the simple tensors e j k . . . e j 1 , where the indices are in weakly monotonic order: 1 j 1 . . . j k . By the standard convention we identify e 0 h ¼ h e 0 ¼ h for any h 2 F WM ðHÞ.
The creation operator M Ã j by the vector e j is defined as follows: The annihilation operator is defined as . . e j 1 ; and they are mutually adjoint: It is useful to introduce the following orthogonal projections. By definition P 0 is the orthogonal projection onto e 0 ¼ X and for m ! 1: Then Q m is the orthogonal projection onto the span of e 0 ¼ X and vectors of the form e j k . . . e j 1 with j k m, and P m is the orthogonal projection onto the span of vectors of the form e j k . . . e j 1 with j k ¼ m, k ! 1. The weakly monotone creation and annihilation operators satisfy the following commutation relations In particular, for j\k we have Remark 12 The monotone creation operators are bounded and generate Ãsubalgebras which are monotone independent in the sense of Muraki [7,8] (for the proof see [13] and [4]).
In this setting the joint monotone numerical radius is defined as and the joint monotone spectral radius of ðT 1 ; . . .; T d Þ is defined as 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 0 :¼ . We also set In particular, for j 2 M 1 and a ¼ ði k ; . . .; i 1 Þ 2 M k we shall have ja ¼ ðj; i k ; . . .; i 1 Þ if j ! a. As one can see for this concatenation 0 plays the role of neutral element and the empty set ; behaves like 0 in multiplication of numbers. Then, with the notation the set fe a : a 2 Mg is an orthonormal basis of F WM ðHÞ. We shall extend this notation to the operators: if T 1 ; . . .; T d 2 BðHÞ are given and a 2 M, then & Now we are ready to describe the explicit formula for the joint monotone numerical radius of d-tuple of operators.
Proposition 14 Let ðT 1 ; . . .; T d Þ be a d-tuple of bounded operators on a Hilbert space H. Their joint monotone numerical radius can be computed as Proof We express h 2 F M ðHÞ H as h ¼ P a2M e a g a with g a 2 H. Then hg a ; T j g ja i; the formula (11) follows. h Note that in the summation over a 2 M the nonzero terms might be only for those a for which ja 6 ¼ ;, i.e. when j ! i k if a ¼ ði k ; . . .; i 1 Þ. For example, if j ¼ 1 then it must be a ¼ 0 or a ¼ ð1; . . .; 1Þ, so in this case the summation over a 2 M 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 1 ; . . .; T d 2 BðHÞ we have Proof Using (7) and (8) and the definition of Q j , we observe that Now, we use the mutual orthogonality of the orthogonal projections P 0 ; P 1 ; . . .; P d to get In analogy to Proposition 6 we have the following properties of the joint monotone numerical radius.
Ad ðiiÞ M . Due to w cl ðTÞ kTk and Lemma 1 we have The other part follows from 1 2 kTk w cl ðTÞ. Ad ðiiiÞ M . Let X : H ! K be a bounded operator and let g 0 ; . . .; 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.
Since the projections Q 1 ; P 2 ; . . .; P d are mutually orthogonal, we obtain Therefore we can write We see that the maximum is achieved for m ¼ 1 and the last expression equals ¼ lim

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 annihilation operators (at first glance it seems to be related to ffiffiffi d p 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.