We investigate mapping properties for the Bargmann transform on modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are well-known when the weights are moderated. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators in the context of these modulation spaces.
The recent theorem by D. Luecking that finite rank Toeplitz-Bergman operators must be generated by a measure consisting of finitely many point masses is carried over to the many-dimensional case.
Theorems about characterization of finite rank Toeplitz operators in Fock-Segal-Bargmann spaces, known previously only for symbols with compact support, are carried over to symbols without that restriction, however with a rather rapid decay at infinity. The proof is based upon a new version of the Stone-Weierstrass approximation theorem.
We study the boundedness of Toeplitz operators on Segal-Bargmann spaces in various contexts. Using Gutzmer's formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal-Bargmann spaces associated to Riemannian symmetric spaces of compact type.
We discuss the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) over the complex $n$-space. This space is characterized by the image of a global Bargmann-type transform introduced by Sjöstrand. We also obtain the deformation estimates of the composition of Berezin-Toeplitz operators whose symbols and their derivatives up to order three are in the Wiener algebra of Sjöstrand. Our method of proofs is based on the pseudodifferential calculus and the heat flow determined by the phase function of the Bargmann transform.
The paper discusses the spectrum of Toeplitz operators in Bargmann spaces. Our Toeplitz operators have real symbols with a variable sign and a compact support. A class of examples is considered where the asymptotics of the eigenvalues of such operators can be computed. These examples show that this asymptotics depends on the geometry of the supports of the positive and negative parts of the symbol. Applications to the perturbed Landau Hamiltonian are given.
We introduce an extended class of cross-Toeplitz operators which act between Fock--Segal--Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-varian…
We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of inhomogeneous quadratic polynomials. It is shown that the boundedness of such operators is implied by the boundedness of the corresponding Weyl symbols.
We deduce continuity, compactness and invariance properties for quasi-Banach Orlicz modulation spaces. We characterize such spaces in terms of Gabor expansions and by their images under the Bargmann transform.
We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of complex quadratic forms, from the point of view of Fourier integral operators in the complex domain. Sufficient conditions are established for the composition of two such operators to be a Toeplitz operator.
The resolvent algebra $\mathcal{R}(X, σ)$ associated to a symplectic space $(X, σ)$ was introduced by D. Buchholz and H. Grundling as a convenient model of the canonical commutation relation (CCR) in quantum mechanics. We first study a representation of $\mathcal{R}(\mathbb{C}^n, σ)$ with the standard symplectic form $σ$ inside the full Toeplitz algebra over the Fock-Bargmann space. We prove that $\mathcal{R}(\mathbb{C}^n, σ)$ itself is a Toeplitz algebra. In the sense of R. Werner's correspondence theory we determine its corresponding shift-invariant and closed space of symbols. Finally, we d…
We establish a characterization of complex linear canonical transformations that are positive with respect to a pair of strictly plurisubharmonic quadratic weights. As an application, we show that the boundedness of a class of Toeplitz operators on the Bargmann space is implied by the boundedness of their Weyl symbols.
We study Toeplitz operators on the Bargmann space, with Toeplitz symbols given by exponentials of complex quadratic forms. We show that the boundedness of the corresponding Weyl symbols is necessary for the boundedness of the operators, thereby completing the proof of the Berger-Coburn conjecture in this case. We also show that the compactness of such Toeplitz operators is equivalent to the vanishing of their Weyl symbols at infinity.
We prove that every multiplier M (bounded operator commuting with the shift operator) on a large class of Banach spaces of sequences on Z is associated to a function essentially bounded by the norm of M on the spectrum of S.
Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L^2-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, we derive explicit polyanalytic sampling theorems which can be seen as a polyanalytic version of the lat…
We look at Toeplitz operators $T_ν$ on the Fock Space (also known as the Segal-Bargmann space) which have a positive Borel measure $ν$ as a symbol. We characterize when $\left(T_ν\right)^s$ for $0<s\leq 1$ is in the symmetrically normed ideal associated with any given symmetric norming function.
We present and study commutative Banach algebras generated by Toeplitz operators with generalized quasi-radial pseudo-homogeneous symbols acting on the Bergman space over the unit ball. We develop the Gelfand theory of these algebras and give some structural information about them. In particular, we provide a description of the radical of these algebras. This paper generalizes and completes the results from previous works related to Toeplitz operators with quasi-radial quasi-homogeneous symbols.
We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.
We describe a construction of wavelets (coherent states) in Banach spaces generated by ``admissible'' group representations. Our main targets are applications in pure mathematics while connections with quantum mechanics are mentioned. As an example we consider operator valued Segal-Bargmann type spaces and the Weyl functional calculus. Keywords: Wavelets, coherent states, Banach spaces, group representations, covariant, contravariant (Wick) symbols, Heisenberg group, Segal-Bargmann spaces, Weyl functional calculus (quantization), second quantization, bosonic field.
In this paper we will give necessary and sufficient conditions for the operator $T_ν^s$ to be in the symmetrically normed ideal $\mathcal{C}_Φ$ for an arbitrary symmetric norming function $Φ$ where $T_ν$ is the Toeplitz operator on the generalized Fock Space (also known as the generalized Bargmann-Fock space) with a positive measure symbol $ν$ and $0<s\leq 1$.
We study the range of time-frequency localization operators acting on modulation spaces and prove a lifting theorem. As an application we also characterize the range of Gabor multipliers, and, in the realm of complex analysis, we characterize the range of certain Toeplitz operators on weighted Bargmann-Fock spaces. The main tools are the construction of canonical isomorphisms between modulation spaces of Hilbert-type and a refined version of the spectral invariance of pseudodifferential operators. On the technical level we prove a new class of inequalities for weighted gamma functions.
The well know conjecture of {\it Coburn} [{\it L.A. Coburn, {On the Berezin-Toeplitz calculus}, Proc. Amer. Math. Soc. 129 (2001) 3331-3338.}] proved by {\it Lo} [{\it M-L. Lo, {The Bargmann Transform and Windowed Fourier Transform}, Integr. equ. oper. theory, 27 (2007), 397-412.}] and {\it Englis} [{\it M. Engli$\check{s}$, Toeplitz Operators and Localization Operators, Trans. Am. Math Society 361 (2009) 1039-1052.}] states that any {\it Gabor-Daubechies} operator with window $ψ$ and symbol ${\bf a}(x,ω)$ quantized on the phase space by a {\it Berezin-Toeplitz} operator with window $Ψ$ and sy…
Following previous works for the unit ball, we define quasi-radial pseudo-homogeneous symbols on the projective space and obtain the corresponding commutativity results for Toeplitz operators. A geometric interpretation of these symbols in terms of moment maps is developed. This leads us to the introduction of a new family of symbols, extended pseudo-homogeneous, that provide larger commutative Banach algebras generated by Toeplitz operators. This family of symbols provides new commutative Banach algebras generated by Toeplitz operators on the unit ball.
Let $Γ$ be a rectifiable Jordan curve, let $X$ and $Y$ be two reflexive Banach function spaces over $Γ$ such that the Cauchy singular integral operator $S$ is bounded on each of them, and let $M(X,Y)$ denote the space of pointwise multipliers from $X$ to $Y$. Consider the Riesz projection $P=(I+S)/2$, the corresponding Hardy type subspaces $PX$ and $PY$, and the Toeplitz operator $T(a):PX\to PY$ defined by $T(a)f=P(af)$ for a symbol $a\in M(X,Y)$. We show that if $X\hookrightarrow Y$ and $a\in M(X,Y)\setminus\{0\}$, then $T(a)\in\mathcal{L}(PX,PY)$ has a trivial kernel in $PX$ or a dense image…
We give criteria for products of Toeplitz and Hankel operators on the Fock (Segal-Bargmann) space to belong to the Dixmier class, and compute their Dixmier trace. At the same time, analogous results for the Weyl pseudodifferential operators are also obtained.
The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain framework for a `maximally possible' extension of the notion of Toeplitz operators for a `maximally wide' class of `highly singular' symbols. Using the language of sesquilinear forms we …
We study mapping properties of Toeplitz operators $T_μ$ associated to nonnegative Borel measure $μ$ on the complex space $\mathbb{C}^n$. We, in particular, describe the bounded and compact operators $T_μ$ acting between Fock spaces in terms of the objects $t$-Berezin transforms, averaging functions, and averaging sequences of their inducing measures $μ$. An asymptotic estimate for the norms of the operators has been also obtained. The results obtained extend a recent work of Z. Hu and X. Lv and fills the remaining gap when both the smallest and largest Banach--Fock spaces are taken into accoun…
We consider Toeplitz and Hankel operators with piecewise continuous generating functions on $l^p$-spaces and the Banach algebra generated by them. The goal of this paper is to provide a transparent symbol calculus for the Fredholm property and a handy formula for the Fredholm index for operators in this algebra.
On the unit ball B^n we consider the weighted Bergman spaces H_λand their Toeplitz operators with bounded symbols. It is known from our previous work that if a closed subgroup H of \widetilde{\SU(n,1)} has a multiplicity-free restriction for the holomorphic discrete series of $\widetilde{\SU(n,1)}$, then the family of Toeplitz operators with H-invariant symbols pairwise commute. In this work we consider the case of maximal abelian subgroups of \widetilde{\SU(n,1)} and provide a detailed proof of the pairwise commutativity of the corresponding Toeplitz operators. To achieve this we explicitly d…
The paper considers bounded linear radial operators on the polyanalytic Fock spaces $\mathcal{F}_n$ and on the true-polyanalytic Fock spaces $\mathcal{F}_{(n)}$. The orthonormal basis of normalized complex Hermite polynomials plays a crucial role in this study; it can be obtained by the orthogonalization of monomials in $z$ and $\overline{z}$. First, using this basis, we decompose the von Neumann algebra of radial operators, acting in $\mathcal{F}_n$, into the direct sum of some matrix algebras, i.e. radial operators are represented as matrix sequences. Secondly, we prove that the radial opera…
On the weakly pseudo-convex domains $Ω_p^n$ we introduce quasi-homogeneous quasi-radial symbols. These are used to prove the existence of a commutative Banach algebra of Toeplitz operators on Bergman space of $Ω_p^n$. We also show that group theoretic and geometric properties for our symbols are satisfied. The results presented here contain the geometric description of the symbols introduced by N. Vasilevski for the unit ball $\mathbb{B}^n$.
We prove a new criterion of weak hypercyclicity of a bounded linear operator on a Banach space. Applying this criterion, we solve few open questions. Namely, we show that if $G$ is a region of $\C$ bounded by a smooth Jordan curve $Γ$ such that $G$ does not meet the unit ball but $Γ$ intersects the unit circle in a non-trivial arc, then $M^*$ is a weakly hypercyclic operator on $H^2(G)$, where $M$ is the multiplication by the argument operator $Mf(z)=zf(z)$. We also prove that if $g$ is a non-constant function from the Hardy space $H^\infty(\D)$ on the unit disk $\D$ such that $g(\D)\cap\D=\va…
We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian theorems are related to localization operators, Toeplitz operators, isomorphism theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization ope…
We give a new proof to show that the complex interpolation for families of Banach spaces is not stable under rearrangement of the given family on the boundary, although, by a result due to Coifman, Cwikel, Rochberg, Sagher and Weiss, it is stable when the latter family takes only 2 values. The non-stability for families taking 3 values was first obtained by Cwikel and Janson. Our method links this problem to the theory of matrix-valued Toeplitz operator and we are able to characterize all the transformations on $\T$ that are invariant for complex interpolation at 0, they are precisely the orig…
For a partition $\boldsymbol{k} = (k_1, \dots, k_m)$ of $n$ consider the group $\mathrm{U}(\boldsymbol{k}) = \mathrm{U}(k_1) \times \dots \times \mathrm{U}(k_m)$ block diagonally embedded in $\mathrm{U}(n)$ and the center $\mathbb{T}^m$ of $\mathrm{U}(\boldsymbol{k})$. We study the Toeplitz operators with $\mathbb{T}^m$-invariant symbols acting on the weighted Bergman spaces on the unit ball $\mathbb{B}^n$. We introduce the $(\boldsymbol{k},j)$-quasi-radial quasi-homogeneous symbols as those that are invariant under the group $\mathrm{U}(\boldsymbol{k},j,\mathbb{T})$ obtained from $\mathrm{U}(…
Let $\mathrm{D}^\mathrm{I}_{n \times n}$ be the Cartan domain of type I which consists of the complex $n \times n$ matrices $Z$ that satisfy $Z^*Z < I_n$. For a symbol $a \in L^\infty(\mathrm{D}^\mathrm{I}_{n \times n})$ we consider three radial-like type conditions: 1) left (right) $\mathrm{U}(n)$-invariant symbols, which can be defined by the condition $a(Z) = a\big((Z^*Z)^\frac{1}{2}\big)$ ($a(Z) = a\big((ZZ^*)^\frac{1}{2}\big)$, respectively), and 2) $\mathrm{U}(n) \times \mathrm{U}(n)$-invariant symbols, which are defined by the condition $a(A^{-1}ZB) = a(Z)$ for every $A, B \in \mathrm{U…
The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L^2(R^d), and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces, modulation spaces and Bargmann-Fock spaces.
We prove sufficient conditions for the boundedness and compactness of Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both the weights $v$ and symbols $a$ are assumed to be radial functions on $\mathbb{D}$. In an earlier work by the authors it was shown that there exists a bounded, harmonic (thus non-radial) symbol $a$ such that $T_a$ is not bounded in any space $H_v^\infty$ with an admissible weight $v$. Here, we show that a mild additional assumption on the logarithmic decay r…
We show that, given a Banach space and a generator of an exponentially stable $C_{0}$-semigroup, a weakly admissible operator $g(A)$ can be defined for any $g$ bounded, analytic function on the left half-plane. This yields an (unbounded) functional calculus. The construction uses a Toeplitz operator and is motivated by system theory. In separable Hilbert spaces, we even get admissibility. Furthermore, it is investigated when a bounded calculus can be guaranteed. For this we introduce the new notion of exact observability by direction. Finally, it is shown that the calculus coincides with one f…
Given a Banach space operator $A$, the isolated eigenvalues $E(A)$ and the poles $Π(A)$ (resp., eigenvalues $E^a(A)$ which are isolated points of the approximate point spectrum and the left ploles $Π^a(A)$) of the spectrum of $A$ satisfy $Π(A)\subseteq E(A)$ (resp., $Π^a(A)\subseteq E^a(A)$), and the reverse inclusion holds if and only if $E(A)$ (resp., $E^a(A)$) has empty intersection with the B-Weyl spectrum (resp., upper B-Weyl spectrum) of $A$. Evidently $Π(A)\subseteq E^a(A)$, but no such inclusion exists for $E(A)$ and $Π^a(A)$. The study of identities $E(A)=Π^a(A)$ and $E^a(A)=Π(A)$, an…
Let $A,$ $T$ and $B$ be bounded linear operators on a Banach space. This paper is concerned mainly with finding some necessary and sufficient conditions for convergence in operator norm of the sequences $\left\{ A^{n}TB^{n}\right\} $ and $\left\{ \frac{1}{n}\sum_{i=0}^{n-1}A^{i}TB^{i}% \right\} $. These results are applied to the Toeplitz, composition and model operators. Some related problems are also discussed.
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<\infty$. Finally, we apply this approach to study the reducing subspaces of $T_{B}$ in weighted Bergman spaces.
The Generalized Locally Toeplitz (GLT) sequences of matrices have been originated from the study of certain partial differential equations. To be more precise, such matrix sequences arise when we numerically approximate some partial differential equations by discretization. The study of the asymptotic spectral behaviour of GLT sequence is very important in analysing the solution of corresponding partial differential equations. The approximating classes of sequences (a.c.s) and the spectral symbols are important notions in this connection. Recently, G. Barbarino obtained some additional results…
