Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov

Abstract We present a survey of some recent results concerning joint numerical ranges of n-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T1, . . ., Tn on a Hilbert space H, there exists a unit vector x ∈ H such that |〈Tjx, x〉| is “large” for all j = 1, . . . , n.


conv σ(T) denotes the convex hull of the spectrum of T. (3) T * = T if and only if W(T) ⊂ R. (4) If T is normal then w(T) = T = r(T) and W(T) = conv σ(T). (5) One has T ≥ w(T) ≥ T . So w(·) is a norm equivalent to · .
The convexity of W(T) is at the heart of the matter of the whole theory of numerical ranges, and underpins many of its directions. The result is due to Toeplitz [82] and Hausdor [47], and has had a number of proofs since then. Three of them can be found in a recent survey [15]. The property ( ) is the rst illustration of the interplay between W(T) and the spectrum of T. There is a plethora of statements of that kind. To give a couple of samples, recall that if T is normal, then every extreme point of W(T) is an eigenvalue of T. In a similar vein, for any T ∈ B(H) a corner point of the boundary ∂W(T) (where a parametrization of W(T) fails to be di erentiable) is an eigenvalue of T as well, see e.g. a discussion in [43, p. 21].
For the sake of completeness we mention also some deeper properties of the numerical ranges. All of them but the last one are elaborated e.g. in [43,Chapter 2]. The striking estimate (2.3) was obtained in [21] developing the ideas from [29]. It led to a number of developments and applications in mathematics including the theory of partial di erential equations and numerical analysis. Some of them are thoroughly presented in [22]. The estimate implies, for instance, that any T ∈ B(H) is similar to an operator having a normal ∂W(T)-dilation, or that T has a so-called skew-normal dilation on

∂W(T) (where ∂W(T) stands for the boundary of W(T).)
For more details on applications of (2.3) to dilation theory we refer to [21], [22] and [75]. A famous open problem posed by Crouzeix is nding the best constant K = K best in (2.3). Crouzeix conjectured that K best = , and, as of now, the best known result is K = + √ . See [23] for the proof of this result, and also the recent papers [9], [19] and [77] for further references, alternative proofs and related statements. There is a number of close works on mapping properties of numerical ranges, resembling ( ) in Theorem 2.3, but concerning a more subtle problem on how numerical ranges behave under appropriate holomorphic maps. For some of the pertinent references one may consult [43], although there are stronger and more recent results. We avoid a discussion of them in this article.
While the spectrum is preserved under similarity transformations, the behavior of W(T) under similarities can be basically arbitrary modulo a constraint in ( ) of Theorem 2.2. The next result clarifying this claim was proved by Williams [84]. The relation (2.4) was obtained by Hildebrandt in [51]. It shows, in particular, that the inclusion in (2.1) is in a sense the best possible. In another paper by Hildebrandt [50], it was shown that conv σ(T) = W(T) if and only if the latter set is the spectral set of T. This further clari es (2.1).
To nish this section, we touch on several issues which are not so popular in the literature, but seem to be important. First, remark that, in general, it is not known which subsets of C can be realized as numerical ranges W(T) of a general T ∈ B(H) or, for example, compact operators on an in nite-dimensional Hilbert space H. The problem seems to be extremely hard and only scattered results are available. For example, the set {z ∈ C : |z| < } ∪ {e iθ : θ irrational} is unattainable as W(T), and a half disc can not be W(T) for a compact T on an in nite-dimensional H. For more on this, including the examples above, see e. g. [1] and [76]. From a slightly di erent geometric perspective, the problem of realization of numerical ranges by operators on C n was studied in [48].
It is also of importance to know which subsets of H may substitute the unit sphere to keep the conclusion of the Toeplitz-Hausdor theorem. Curiously, this question has essentially escaped the attention of experts. A deep study of that problem was initiated in [35]. In particular, it was proved in [35, Theorem 2.1 and Theorem 4.3] that the unit sphere in H can be replaced by a closed annulus {x ∈ H : a ≤ x ≤ b} for any b ≥ a ≥ . Moreover, by [35,Theorem 5.10], if T is selfadjoint, then the unit sphere can be replaced by a "slice" {x ∈ H, x = : a ≤ Tx, x ≤ b} for any a, b ∈ R, a ≤ b.
In another interesting paper [41], the authors showed that for T ∈ B(C n ) one can introduce a so-called numerical measure µ T on W(T) being a push-forward of the Haar measure on the unit sphere in C n under the map x → Tx, x . The support of µ T coincides with W(T), and it seems that the numerical measure µ T captures a lot of information on W(T). Such an approach has non-trivial applications to the study of partial di erential equations. However, it seems, it did not attract the attention it deserves.
The theory of numerical ranges is intimately related to optimization theory and convex analysis. However, the interconnections between these subjects seem to be not exploited enough by the operator-theoretical community. For sample papers discussing a number of them, see e.g. [52] and [31].
There are hundreds of papers on numerical ranges, their properties and various generalizations, including descriptions of numerical ranges for particular classes of operators. However, while for some classes, like Toeplitz operators, a detailed information is available, the other (even quite close) classes, such as Hankel operators, have not received an adequate treatment. We are only aware of an old paper [83] treating the numerical ranges of Hankel operators among other things. Thus, a lot is still to be done.

Joint numerical ranges, joint essential numerical ranges and their relatives
For an n-tuple T = (T , . . . , Tn) ∈ B(H) n we de ne the joint numerical range as The set W(T) can be identi ed with a subset of R n if one identi es the n-tuple T with the ( n)-tuple (Re T , Im T , ..., Re Tn , Im Tn) of selfadjoint operators, where the real (imaginary) part of an operator T ∈ B(H) is de ned by Re T = T+T * and Im T = T−T * i . Such an identi cation is often useful, however certain statements require additional care.
To deal with operator tuples, it would be convenient to introduce the following notation. For x, y ∈ H and n ∈ N we will be writing Tx, y = ( T x, y , . . . , Tn x, y ) ∈ C n and Tx = (T x, . . . , Tn x) ∈ H n .
The non-commutativity of operator entries of T makes the study of W(T) much more demanding than in the setting of a single operator, and it leads to a number of new geometric phenomena. In particular, the joint numerical range of an n-tuple of operators is, in general, not convex if n ≥ , as the next example shows. Hence W(T, S) is not convex.
In the example above, T is selfadjoint. So the triple (T, Re S, Im S) is an example of a triple of selfadjoint operators with non-convex numerical range. Such an example exists only in two-dimensional spaces. We discuss this phenomenon below. However, in any space there exist pairs of operators (or equivalently, -tuples of selfadjoint operators) with non-convex numerical range.
As in the previous example one has which is not convex. Hence W(T, S) is not convex.
Apparently, the fact that the joint numerical range may fail to be convex was known already to Hausdor [47]. Various variations of the above examples were given in [8], [61] and [34, p. 33-34] (where it is claimed that a similar example was produced by Halmos). The n-tuples of operators of the form (T, T , . . . , T n ), where T is a xed operator T ∈ B(H), are of particular interest since they allow one to link the theory of joint numerical ranges to ne properties of T. Such tuples will be of primary importance in the next section. Note that the geometry of the corresponding numerical ranges W(T, . . . , T n ) is still far from being understood. In particular, we do not know whether W(T, . . . , T n ) is always convex if n ≥ .
While W(T) is not convex in general, it still posses some traces of convexity. As shown in [27], with any two points it contains an ellipsoid (perhaps degenerate) joining them. See also Theorem 3.6 below.
There are several instances when W(T , . . . , Tn) is convex. For example, W(T , . . . , Tn) is convex if n = and T j , ≤ j ≤ , are selfadjoint and dim H ≥ , see [45]. This leads, in particular, to the following curious observation made in [62]. The set DW(T) is called the Davis-Wielandt shell in the literature. Clearly, its geometry provides more information about T than the usual numerical range W(T), which is just a projection of DW(T) on the rst coordinate. In particular, the normality of T ∈ B(C n ) can be described solely in terms of DW(T). For more details on DW(T) one may consult [62] or [65] and the references cited therein.
An interesting illustration of the interplay between Tx, x and Tx for x ∈ H, and thus the structure of DW(T), is provided by Garske's theorem, [42]. It says that if T ∈ B(H), then where R is the radius of the smallest disk containing the spectrum of T. Moreover, as shown in [12], if T is normal, then the above inequality becomes equality. For a tuples version of this result see Section 4.
The notion of the Davis-Wielandt shell is closely related to the notion of maximal numerical range, which has been also studied in the literature. Among the very rst papers in this direction, we mention [80] and [38].
We note that for a pair of bounded operators (T , T ) ∈ B(H) another joint numerical range W(T , T ) is introduced in the apparently forgotten paper [7]. In particular, while W(T , T ) is not in general convex, it was proved in [7] that the set W(T , Another instance when the convexity of W(T , ..., Tn) still survives, arises when the T j have a certain special algebraic structure. As a simple illustration one may recall that W(T , ..., Tn) is convex if T j , ≤ j ≤ n, are (bounded) commuting normal operators on H, [24]. Here the assumption of commutativity is essential. Without commutativity it is no longer true.
Finally, we mention yet another situation when the joint numerical range is convex. Let H be a separable in nite-dimensional Hilbert space. Denote by S (H) the set of all Hilbert-Schmidt operators on H, and by S (H) the set of all trace-class operators on H. Then S (H) with the inner product given by X, Y S = trace (Y * X), X, Y ∈ S , is again a Hilbert space, while S a Banach space with the norm de ned as X S = trace(X * X) / , X ∈ S (H). We will return to S (H), S (H) and similar classes in Section 5.3. For T ∈ B(H) denote by T : S (H) → S (H) the operator de ned by T(X) = TX, X ∈ S (H). As for subsets of C n , if S ⊂ C n , then the convex hull of S is denoted by conv S. The next statement seems to be new. Clearly λ ∈ conv W(T). Hence W( T) ⊂ conv W(T).
We show now that conv W(T) ⊂ W( T). Let µ ∈ conv W(T) ⊂ C n and we will identify further C n with R n . Then µ is a convex combination of at most n + elements of W(T), i.e., The assumption Y S = trace(Y) = implies that Y ≥ and that Y can be represented as Before passing to the next main object of our studies, we mention that the following problem seems to be open. Recall that for S ⊂ C n its polynomial (convex) hullŜ is de ned aŝ Accordingly, S is polynomially convex if S =Ŝ. Similarly to the spectral theory of linear operators, there is an "essential version" of the notion of the joint numerical range. It will play the central role in our subsequent considerations. Let dim H = ∞ and T = (T , . . . , Tn) ∈ B(H) n . We de ne the joint essential numerical range We(T) of T as the set of all n-tuples λ = (λ , . . . , λn) ∈ C n such that there exists an orthonormal sequence ( The joint essential numerical range We(T) can be also de ned as where the intersection is taken over all n-tuples (K , . . . , Kn) of compact operators on H. For the proof of the equivalence of these de nitions one may consult [70,Theorem 2]. For n = , the essential numerical range was introduced and studied in depth in the in uential paper [36]. A Banach algebra counterpart of the essential numerical range was de ned earlier in [79].
Being an approximate version of W(T), the set We(T) appeared to be better adopted to the use of spectral theory for T, and its invariance under compact perturbations illustrates the speci cs of We(T) very well. Moreover, in contrast to W(T), the joint essential numerical range is always convex. Some of the geometric properties of We(T) are summarized in the next result taken from [63]. Recall that a set S ⊂ C n is star-shaped if there is a point in S, called a star center, that can be connected by a line segment with any other point in S. It is known that a star-shaped set is simply connected.

Theorem 3.6. Let T ∈ B(H) n . Then We(T) is a compact convex subset of the star-shaped set W(T). Moreover, each element in We(T) is a star center of W(T).
The analogy to spectral theory mentioned above may serve as a good intuition, however the corresponding relations between W(T) and We(T) are more involved than those of their spectral counterparts. Note that for

T ∈ B(H) one may have We(T) ∩ W(T) = ∅ and points from W(T) ∩ We(T) may not be star-centers for W(T).
Moreover, for any n ≥ , the set W(T) may be not convex even if W(T) is convex, see [63]. The di erence between W(T) and We(T) is illustrated by the fact that while realizing convex sets by numerical ranges is a hard open problem, for any compact convex set S ⊂ C n there exist n-tuples T and T from B(H) n such that S = We(T ) = W(T ), see [63,Corollary 5.4].
Apparently the convexity of We(T , . . . , Tn) was rst proved in [11, Lemma 3.1] (where even a more general result can be found). See also [63] for a di erent proof and further penetrating study of We(T), including its geometric properties, stability under perturbations, examples, etc.
Let us present yet another argument yielding the convexity of We(T) and based on the next simple but useful observation.
Note that if T = (T , . . . , Tn) ∈ B(H) n , λ ∈ C n belongs to We(T) if and only if for every δ > and every subspace M ⊂ H of nite codimension there exists a unit vector x ∈ M such that || Tx, x − λ|| C n < δ. The observation was used without proof in [72] and the proof of its non-trivial implication was given in [73,Lemma 4.1], see also [73,Proposition 5.5]). We justify the "only if" implication, and omit the other implication whose proof is straightforward. To this aim, note that if λ ∈ We(T), then there exists an orthonormal sequence ( Proof. Either of the two equivalent de nitions of We(T) implies immediately that We(T) is a closed set.
To prove the convexity, let λ, µ ∈ We(T) and t ∈ [ , ]. Assume that M ⊂ H is a subspace ofnite codimension. By the observation above, there exists an orthonormal sequence (x k ) ∞ k= in M such that lim k→∞ Tx k , x k = λ. Similarly, we can construct inductively an orthonormal sequence (y k ) ∞ k= such that for every k ∈ N, Hence tλ + ( − t)µ ∈ We(T).
Thus, the joint essential numerical range We(T) has better geometric properties than the joint numerical range W(T). On the other hand, the joint numerical range W(T) provides more information about the n-tuple T = (T , . . . , Tn), and is more explicit. By mere de nitions, see [73,Theorem 5.1] and the discussion preceding it. This is a generalization for operator tuples of the famous theorem due to Lancaster for n = .
Lancaster's theorem yields quite useful results, e.g. descriptions of the situations, when W(T) is closed or open. Nevertheless, the presence of W(T) on both sides of (3.1) makes the equality somewhat implicit. So, it is a natural question which part of We(T) is contained in W(T). The next theorem proved in [73,Corollary 4.2] provides a partial answer. For S ⊂ C n denote by Int S its topological interior.
As a consequence, we can nd a joint diagonal compression for T , . . . , Tn to an in nite-dimensional subspace of H, see [72,Corollary 4.3]. Recall that for T ∈ B(H) the problem of characterizing λ ∈ C such that P T P = λP for an in nite rank projection P was posed in [36, p. 190]. The special case of n = of the following statement was proved in [3, p. 440].
where P L is the orthogonal projection on L.
Proof. By Theorem 3.8, there exists a unit vector x ∈ H such that Tx , x = λ. Construct inductively a sequence (x k ) ∞ k= ⊂ H of unit vectors such that Then, in view of our construction of (x k ) ∞ k= , it is easy to see that Ty, y = λ||y|| . Since the choice of y is arbitrary, P L T j P L = λ j P L , for all ≤ j ≤ n.
Closely related to the notion of joint essential numerical range are higher rank numerical ranges. These numerical ranges have been studied intensively, e.g. in connection with the quantum computing, see [64] and the references therein.
Let T = (T , . . . , Tn) ∈ B(H) n and ≤ k ≤ ∞. We de ne the k-th rank numerical range W k (T) of T as the set of all (λ , . . . , λn) ∈ C n such that there exists a subspace L ⊂ H, with dim L = k satisfying P L T j P L = λ j P L , j = , . . . , n.
(Note that W k (T) are usually denoted by Λ k (T) in the literature, while the notation W k (T) is used for so-called k-th numerical ranges. However, we preferred the more intuitive notation above.) Observe that W (T) is the usual joint numerical range and For k ∈ N, the set W k (T) is, in general, not convex, but it is always non-empty and star shaped. At the same time, it is easy to see that the in nite numerical range W∞(T) can be empty even if n = (by considering an injective positive de nite compact operator T ), but W∞(T) is always convex. In a sense, We(T) is an approximate version of W∞(T). This is summarized in the theorem below.   (t n− Re T j + t n Im T j ) + cI is compact, then t = · · · = t n = .
Having an approximate character, essential numerical ranges and sets of similar nature are traditionally expressed as intersections of numerical ranges of tuples over appropriate classes of perturbations, usually compact ones. Somewhat surprisingly, in the following result obtained in [73,Theorem 5.8], We(T) is described by means of unions of the in nite numerical ranges of compact perturbations T + K, where K is an n-tuple of compact operators.

Joint numerical ranges and spectrum
As it was mentioned in Theorem 2.2 (2), conv σ(T) ⊂ W(T) for each single operator T ∈ B(H). Unfortunately, for non-commuting tuples there is no convenient joint spectrum, although the notion of joint numerical range can be de ned properly. On the other hand, for commuting tuples there are many, comparatively useful de nitions of spectrum (Taylor, Harte, approximate point spectrum, surjective spectrum, ...), which, in general, may di er from each other. However, all reasonable spectra in this setting have the same convex hull, [69,Chapter III].
For de nitiveness, we assume below that for T ∈ B(H) n , the notation σ(T) stands for the Harte spectrum of T. The next theorem due to V. Wrobel is a version of Theorem 2.    Note in passing that there is a partial analogue of Theorem 2.4 for operator tuples, obtained in [37]. For commuting operators T , . . . , Tn and for every j, ≤ j ≤ n, let S j be an open convex set containing σ(T j ). Then there exists an invertible operator R such that W(RT j R − ) ⊂ S j for every j. The commutativity assumption cannot be removed here. The case n = was considered in Theorem 2.4, (i). However the description of conv σ(T) in terms of similarities as in Theorem 2.4 is apparently missing in the literature.
It is curious to note that Garske's theorem mentioned in Section 3, generalizes to tuples of operators. As proved in [33], if T , . . . , Tn ∈ B(H) are mutually commuting, then where R is the radius of the smallest ball containing the (Harte) joint spectrum of (T , . . . , Tn). Moreover, the above inequality becomes equality if T j , ≤ j ≤ n, are mutually commuting normal operators on H.
The joint numerical ranges W(T, . . . , T n ) of powers of a single operator have certainly their own speci cs. The next consequence of Corollary 4.3 proved in [72,Theorem 4.6] is instrumental in all of our applications of the theory of joint numerical ranges. Note that it allows one to deal with the polynomial hullσ(T) ⊂ C rather than the much less transparent set conv σ(T, ..., T n ) ⊂ C n . Recall thatσ(T) can be described as the union of σ(T) with all bounded components of the complement C \ σ(T).

Several applications of joint numerical ranges
Now we turn to several applications of joint numerical ranges to other problems in operator theory found recently in [72], [73], and [74]. The general ideology developed in [72]- [74] is that to every T ∈ B(H) one associates an n-tuple Tn := (T, ..., T n ), and tries to uncover ne properties of T in terms of the structure of the sets σ(T), W(Tn), and We(Tn), n ∈ N, rather than a single set W(T). While the study of an operator T in terms of asymptotic or algebraic properties of its powers is a rather standard approach going back to the birth of operator theory, this idea of invoking the sequences of numerical ranges W(Tn) and We(Tn) had not been exploited until recent time.
Apart from the papers [72]- [74] developing the approach above, one may mention [27], where very particular results on W(T, ..., T n ) were obtained for T ∈ B(C n ).
In this section, we present various generalizations and improvements of notorious operator-theoretical results accomplished by using numerical ranges techniques.

. Circles in the spectrum
First, we characterize the circle structure in the spectrum of a bounded linear operator linking in this manner several statements from ergodic theory, harmonic analysis and spectral theory. We start with an old and elegant theorem of W. Arveson proved in his PhD thesis, see [4]. Among the motivations for the result, there is a classical Rokhlin Lemma for measure preserving transformations, one of the building blocks of ergodic theory. The result can be considered a spatial version of the lemma. Note that the second condition in the above theorem can be reformulated as ( , . . . , ) ∈ W(U, U , . . . , U n ) for every n ∈ N. This suggests the use of numerical ranges for tuples and motivates our studies in Section 4. In view of the results in Section 4, the implication (i)⇒(ii) can be generalized as follows. Proof. From Theorem 4.4, we have that ( , . . . , ) ∈ Int We(T, T , . . . , T n ) and Corollary 3.9 concludes the proof.
Since for unitary operators T one has σ(T) ⊂ T, the above corollary can be further sharpened.

Theorem 5.3. Let U ∈ B(H) be a unitary operator. The following statements are equivalent: (i) σ(U) = T;
(ii) for every n ∈ N there exists an in nite-dimensional subspace L ⊂ H such that P L U j P L = , j = , . . . , n.
The ultimate general form of Arveson's Theorem 5.1 seems to be the following statement, [72, Theorem 1.1].

Theorem 5.4. Let T be a bounded linear operator on H, such that the spectral radius r(T) ≤ . The following statements are equivalent. (i) T ⊂ σ(T). (ii) For all ϵ > and n ∈ N there exists x ∈ H such that
(iii) For all ϵ > and n ∈ N there exists x ∈ H such that

. Numerical ranges and asymptotics of weak orbits
Another motivation for the study of the circle structure of the spectrum stems from an interplay of ergodic theory and harmonic analysis. Recall that a positive measure ν on the unit circle T is called Rajchman if its Fourier coe cients ((Fν)(n)) n∈Z satisfy (Fν)(n) → , |n| → ∞. While this class of measures is crucial in many chapters of analysis and appears frequently in the literature, no handy characterization of it is available so far, see e.g. [68] for discussions of results and problems behind it. In his studies of weak mixing properties of dynamical systems, D. Hamdan proved in [46] that if ν is Rajchman, then supp ν = T if and only if for every ϵ > there exists a positive f ∈ L (T, ν) such that ν-Fourier coe cients of f given by (Fν f )(n) = T z n f (z) dν(z), n ∈ Z, are uniformly small in the sense that sup n∈Z |(Fν f )(n)| < ϵ. Note that if (Uf )(z) = zf (z), then U is unitary on L (T, ν), U n → in the weak operator topology, and σ(U) = supp ν. This operator-theoretical interpretation of the Hamdan's result on Rajchman measures leads to the following theorem proved in [46] for unitary operators induced by measure preserving transformations.  [72] that one is allowed to take elements x in (ii) from a speci ed in nite-dimensional subspace. Namely, the following result was obtained.

Theorem 5.6. Let T ∈ B(H), and let T n → in the weak operator topology. Suppose that ∈ Intσ(T). Then for every ε > there exists an in nite-dimensional subspace L of H such that
sup n≥ P L T n P L ≤ ε and lim n→∞ P L T n P L = .

In particular, this is true if the assumption ∈ Intσ(T) is replaced by T ⊂ σ(T).
If T is unitary then the statement above can be improved. The following result generalizes Theorem 5.5 (by using a completely di erent approach than that of [46]). For its proof see [72,Corollary 6.5].

Corollary 5.7. Let T be a unitary operator on H such that T n → in the weak operator topology. Then any of the conditions (i) and (ii) of Theorem 5.5 is equivalent to the condition (iii) For every ε > there exists an in nite-dimensional subspace L ⊂ H such that
sup n≥ P L T n P L ≤ ε and lim n→∞ P L T n P L = . .

Diagonals of operators
Let T ∈ B(H). Assume for de nitiveness that H is an in nite-dimensional separable Hilbert space. The problem we address in this section is how to describe all possible diagonals of T, i.e., all sequences (d k ) ∞ k= such that d k = Tu k , u k for all k ∈ N and some orthonormal basis (u k ) ∞ k= in H. The problem appears naturally in many situations and has been studied intensively. A related, second problem is how to describe all possible diagonals of operators in a given class. These problems are naturally connected with the numerical range and its subsets, since the entries constituting diagonals of T belong to W(T). For a detailed discussion of some motivations for such studies we refer to the introduction in [74]. Here we just quote a claim from [32] speculating that "the diagonal of an operator carries more information about the operator than its relatively small size (compared to the "fat" matrix representation of the operator) may suggest." Most of the research on diagonals is concentrated on the second problem. Answering a question by A. Gillespie, C. K. Fong showed that in [39] that for any (d k ) ∞ k= ∈ ∞ there exists T ∈ B(H), T = such that (d k ) ∞ k= is a diagonal of T. (As it was remarked later by Herrero in [49,Section 4], the exponent can be replaced by .) The proof was inspired by a deep characterization of operators possessing a zero diagonal, obtained in [32, Theorem 1] by P. Fan. (There was a aw in Fan's argument which was corrected recently in [67].) While Fong's result led to further research on diagonals, it remained essentially the only result of this kind for a long while.
In the beginning of this century, being motivated by problems from the theory of C * -algebras, R. V. Kadison described in [55,56] the diagonals for a class of selfadjoint projections on H. Kadison's elegant result can be stated as follows.
Theorem 5.8 . (Note that the situation changes dramatically if one drops the orthogonality assumption. The set of diagonals of idempotents on H then lls the whole of ∞ , see [66].) Almost immediately, W. Arveson extended in [6] the realm of Kadison's considerations to normal operators with nite spectrum, see also [5]. Kadison-Arveson's perspective generated an activity on characterizing the set of diagonals for several classes of operators: selfadjoint, unitary or normal under various spectral assumptions, and gave rise to a number of deep results. As a sample we mention the next recent characterization of diagonals for a class of unitary operators, see [53].
The descriptions of diagonals for operator classes became a part of a long research program realized by Bownik, Jasper, Kaftal, Loreaux, Weiss, and others. For some of their achievements, see [17], [18], [57], [58], [66], [53] and the citations in these papers. There is also a separate and similar direction in the setting of C * -algebras. We omit a discussion of it and refer e.g. to [59], [60] and the references therein.
An inspiration to our studies was the paper [49] by D. Herrero, motivated in part by [39] and addressing a more demanding problem of description of diagonals for a xed operator. Let us rst introduce some notation.
The theorem admits a slightly more general formulation involving tuples of selfadjoint operators on H. Note that for a selfadjoint operator T ∈ B(H) the spectrum of T may contain an interval, but the interior of W(T) could be empty in this case. So, in order not to miss several situations of interest, one should deal with the notion of relative interior. For more details see [74]. If n = and We(T) coincides with the closed unit disc D, then the assumption ∞ k= dist {d k , ∂We(T)} = ∞ reduces to the negation of the classical Blaschke condition ∞ k= ( − |d k |) = ∞, and this explains our terminology.
The Blaschke-type assumption in Theorem 5.11 is, in some sense, the best possible as the next example from [49] shows. Note that to formulate and to prove such a result for tuples of powers one has to invoke the polynomial hull of σ(T), rather than We(T). Its proof can be found in [74,Corollary 4.11]. So the statement follows from Theorem 5.11.
An interesting interplay between the assumption ∈ We(T) and the structure of D(T) was discovered by Q. Stout in his studies of Schur algebras. These are commutative Banach algebras of in nite matrices de ned by Schur multiplication, i.e., the term-wise product of the matrix representations of operators of a Hilbert space (given an orthonormal basis). Q. Stout in [81] proved that the condition of zero belonging to the essential numerical range of T ∈ B(H) is equivalent to several properties revealing the structure of a Schur algebra. The next result, due to Q. Stout ([81, Theorem 2.3]), relates the essential numerical range of T to its diagonals.
Theorem 5.15 (Stout, 1981 This theorem generalizes an older result of J. Anderson that arises in the study of commutators of operators. Recall that ∈ We(T) if and only if there exists (d k ) ∞ k= in D(T) such that (d k ) ∞ k= belongs to c (N). Anderson proved that ∈ We(T) is in fact equivalent to the existence of a p-summable sequence in D(T) for every p > . Corollary 5.16 (Anderson, 1971). Let T ∈ B(H), ∈ We(T) and p > . Then there exists an orthonormal basis A similar statement (with a di erent proof) was used in [40,Theorem 4.1] Note that the following question of Stout, related to Theorem 5.15, seems to be not yet answered. Given an operator T ∈ B(H) such that ∈ We(T), and a sequence (an) ∞ n= of positive numbers which is not in , does there exist a basis (en) ∞ n= and a bijection π from N × N onto N such that | Ten , em | < a π(n,m) for all n and m ? (Apparently, there is a misprint in the formulation of this question in [81].) It was also asked in [81]   consisting of constant diagonals. Understanding the structure of D const (T) for a xed T and relating it to the structure of We(T) was a natural next step. Clearly we have that By Theorem 3.7, the set We(T) is convex, and since the interior of a convex set is convex, so is Int We(T). However, the question whether D const (T) is convex is still open, although we have a positive answer if n = (unpublished). This problem has been raised by J.-C. Bourin in [16].

. Block operator diagonals
A natural generalization of diagonals are block diagonals. Block diagonals arise in a variety of issues from operator theory, ranging from the study of quasitriangularity and quasidiagonality to the investigations of unitary and similarity orbits and their spans. Being unable even to touch them, we mention the paper [26] as a sample, where block diagonals and numerical ranges appeared to be crucial in the latter circle of problems. Note that the block diagonals are sometimes called "pinchings" in the literature.
As we will see below, the study of block diagonals is intimately related to essential numerical ranges. However, besides the numerical ranges structure, one has to use new operator-theoretical constructions somewhat similar to dilations. Their description however falls out of the scope of this survey, and we refer to [74] for more explanations and details.
The following result was proved in [16].
Theorem 5.19 (Bourin 2003). Let T ∈ B(H) with We(T) ⊃ D. Let L k , k ∈ N, be separable Hilbert spaces ( nite or in nite-dimensional), and let C k ∈ B(L k ) be contractions satisfying sup k C k < . Then there exist projections P K k , k ∈ N, onto mutually orthogonal subspaces K k ⊂ H such that ∞ k= K k = H and P K k TP K k is unitarily equivalent to C k , for all k ∈ N.
In [74], we extended Theorem 5.19 to the setting of tuples and replaced the uniform contractivity condition on the operator diagonal by a more general assumption of Blaschke's type, see [74,Theorem 1.3]. Such an assumption is, in general, necessary even for scalar diagonals as Example 5.12 shows.
Theorem 5.20. Let T ∈ B(H) with We(T) ⊃ D. Let L k , k ∈ N, be separable Hilbert spaces ( nite or in nitedimensional) and let C k ∈ B(L k ) be strict contractions satisfying ∞ k= ( − ||C k ||) = ∞. Then there exist projections P K k , k ∈ N, onto mutually orthogonal subspaces K k ⊂ H such that H = ∞ k= K k , and P K k T| K k is unitarily equivalent to C k , for all k ∈ N.
Replacing the numerical range condition We(T) ⊃ D in Theorem 5.20 by the spectral assumption σ(T) ⊃ D, we can put Theorem 5.20 in a more demanding context of tuples of powers of T. For a sequence of Hilbert space contractions (C k ) ∞ k= with norms not approaching too fast, the following statement, proved in [74, Theorem 6.3], yields pinchings (C k , . . . , C n k ) for a tuple (T, . . . , Tn), T ∈ B(H), if the spectrum of T is su ciently large. It would be instructive to note the analogy to Theorem 5.14.
Theorem 5.21. Let T ∈ B(H), σ(T) ⊃ D, n ∈ N. Let L k , k ∈ N, be separable Hilbert spaces, and let C k ∈ B(L k ), k ∈ N, be strict contractions such that ∞ k= ( − C k ) n = ∞. Then there are mutually orthogonal subspaces K k , k ∈ N, of H such that H = ∞ k= K k , and P K k (T, . . . , T n )P K k is unitarily equivalent to (C k , . . . , C n k ) (in an entry-wise sense) for all k ∈ N.

On joint numerical radius
All results of this section are contained in [71] and [30]. In particular, Proposition 6.3 and Theorem 6.4 originate from [71], while Theorems 6.6, 6.7 and 6.8 were obtained in [30].
Recall that one of the basic properties of the numerical radius is the inequality for all operators T ∈ B(H). Equivalently, for every ε > there exists x ∈ H, x = such that If, moreover, dim H < ∞, then there exists x ∈ H, x = , such that | Tx, x | ≥ T . In this section we will discuss an analogous property for n-tuples of operators. We consider the following problem: Tn ∈ B(H). Does there exist x ∈ H, x = , such that | T j x, x | is "large" for all j = , . . . , n?
The problem is closely related to the so-called Tarski's plank problem on covering a convex body in R n by strips, its solution by Bang and further developments by Ball and others. For a nice review of this subject, one may consult the survey [10].
We start with a couple of useful reductions. First, dealing with Problem 6.1, one may assume that dim H < ∞. Indeed, it su ces to note that for (T , ..., Tn) ∈ B(H) n one has W(T , ..., Tn) = P W(PT P, ..., PTn P), where P runs over all nite-rank orthogonal projections (or merely over orthogonal projections of rank not exceeding n + ). Second, replacing the operators T j by their real and imaginary parts Re T j and Im T j , and noting that | T j x, x | ≥ max | Re T j x, x |, | Im T j x, x | , x ∈ H, we may consider (without much loss of generality) only tuples of selfadjoint operators. Thus we may study the following reformulation of Problem 6.1: If the operators T j are not only selfadjoint, but also positive semi-de nite, then one can obtain the next precise answer. The constant /n is the best.
For selfadjoint operators T j , ≤ j ≤ n, the exact estimate is known only for n = and n = . A positive answer to this problem would allow us to set cn = n− in the estimate in Problem 6.2. It is known that the answer is indeed positive for n = and n = , and this is used in the proof of Theorem 6.4. As remarked in [71,Example 8], the estimate n− in Problem 6.5 cannot be improved. Indeed, let n ∈ N and let u j = (u j , . . . , u jn ) ∈ R n be de ned by u jj = , j = , . . . , n, and u jk = − n− , for ≤ j, k ≤ n, j ≠ k.
If v = (v , . . . , vn) is an arbitrary vector from the convex hull of {u , . . . un}, then min ≤k≤n |v k | ≤ n− . In general, for n ≥ , it is only known that there exits y ∈ Q from the convex hull of x (k) , ≤ k ≤ n, with |y k | ≥ n √ n for all k. The last property can be applied in all situations where an appropriate numerical range of (T , . . . , Tn) is convex. Apart from the situations discussed in Section 3, we mention the following set-up. Let A be a unital Banach algebra, and let a , . . . , an ∈ A. De ne the algebraic numerical range V(a , . . . , an , A) = (f (a ), . . . , f (an)) : f ∈ A * , f = = f ( A ) .
Thus the partial answer to Problem 6.2 leads, for instance, to the following results. One can also prove an asymptotic version of the estimate treated in this section. The best known estimate for the joint numerical range of general operators is the following result. So for the constant cn in Problem 6.2, there is still a large gap between the plausible upper estimate n− (veried in several particular cases) and the lower estimate n . We conjecture that cn should be proportional to n .