This paper is concerned with polynomially generated multiplier invariant subspaces of the weighted Bergman space $A_{\boldsymbolβ}^2$ in infinitely many variables. We completely classify these invariant subspaces under the unitary equivalence. Our results not only cover cases of both the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$ and the Bergman space $A^{2}(\mathbb{D}_{2}^{\infty})$ in infinitely many variables, but also apply in finite-variable setting.
In a wide class of weighted Bergman spaces, we construct invertible non-cyclic elements. These are then used to produce z-invariant subspaces of index higher than one. In addition, these elements generate nontrivial bilaterally invariant subspaces in anti-symmetrically weighted Hilbert spaces of sequences.
Let $\mathcal{W}$ be the corresponding wandering subspace of an invariant subspace of the Bergman shift. By identifying the Bergman space with $H^2(\mathbb{D}^2)\ominus[z-w]$, a sufficient and necessary conditions of a closed subspace of $H^2(\mathbb{D}^2)\ominus[z-w]$ to be a wandering subspace of an invariant subspace is given also, and a functional charaterization and a coefficient characterization for a function in a wandering subspace are given. As a byproduct, we proved that for two invariant subspaces $\mathcal{M}$, $\mathcal{N}$ with $\mathcal{M}\supsetneq\mathcal{N}$ and $dim(\mathcal…
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.
We answer affirmatively the problem left open in \cite{DSZ,GSZZ} and prove that for a finite Blaschke product $ϕ$, the minimal reducing subspaces of the Bergman space multiplier $M_ϕ$ are pairwise orthogonal and their number is equal to the number $q$ of connected components of the Riemann surface of $ϕ^{-1}\circ ϕ$. In particular, the double commutant $\{M_ϕ,M_ϕ^\ast\}'$ is abelian of dimension $q$. An analytic/arithmetic description of the minimal reducing subspaces of $M_ϕ$ is also provided, along with a list of all possible cases in degree of $ϕ$ equal to eight.
A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szegö and the Bergman kernel on the symmetrized polydisc.
Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc $\mathbb{D}^n$ (with $ n \geq 2$) is investigated. We show that for any non-empty subset $α=\{α_1,\dots,α_k\}$ of $\{1,\dots,n\}$ and doubly commuting invariant subspace $\s$ of the Bergman space or the Dirichlet space over $\D^n$, the tuple consists of restrictions of co-ordinate multiplication operators $M_α|_\s:=(M_{z_{α_1}}|_\s,\dots, M_{z_{α_k}}|_\s)$ always possesses wandering subspace of the form \[\bigcap_{i=1}^k(\s\ominus z_{α_i}\s). \]
It is well known that the structure of nontrival invariant subspaces for the shift operator on the Bergman space is extremely complicated and very little is known about their specific structures, and that a complete description of the structure seems unlikely. In this paper, we find that any invariant subspace $M(\neq \{0\})$ for the shift operator $M_z$ on the Bergman space $A^{^p}_α(\D)~(1\leq p< \infty,~~-1<α<\infty)$ contains a nonempty subset that lies in $A^{^p}_α(\D)\backslash H^{^p}(\D)$. To a certain extent, this result describes the specific structure of every nonzero invariant subsp…
In this note we study some basic properties of general fractional derivatives induced by weighted Bergman kernels. As an application we demonstrate a method for generating pre-images of analytic functions under weighted Bergman projections. This approach is useful for proving the surjectivity of weighted Bergman projections in cases when the target space is not a subspace of the domain space (such situations arise often when dealing with Bloch and Besov spaces). We also discuss a fractional Littlewood-Paley formula.
In this paper, we show that on the weighted Bergman space of the unit disk the essential norm of a noncompact Hankel operator equals its distance to the set of compact Hankel operators and is realized by infinitely many compact Hankel operators, which is analogous to the theorem of Axler, Berg, Jewell and Shields on the Hardy space in Axler et al. (1979); moreover, the distance is realized by infinitely many compact Hankel operators with symbols continuous on the closure of the unit disk and vanishing on the unit circle.
This paper is devoted to the study of reducing subspaces for multiplication operator $M_ϕ$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_ϕ$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_ϕ$ on the Bergman space, and we discover a new way to study the Riemann surface for $ϕ^{-1}\circϕ$. By this means, we determine the reducing subspaces of $M_ϕ$ on the Dirichlet space when the order of $ϕ$ is $5$; $6$; $7$ and answer some questions of Douglas-Putin…
Following upon results of Putinar, Sun, Wang, Zheng and the first author, we provide models for the restrictions of the multiplication by a finite Balschke product on the Bergman space in the unit disc to its reducing subspaces. The models involve a generalization of the notion of bundle shift on the Hardy space introduced by Abrahamse and the first author to the Bergman space. We develop generalized bundle shifts on more general domains. While the characterization of the bundle shift is rather explicit, we have not been able to obtain all the earlier results appeared, in particular, the facts…
In this paper, we use a new method to solve a long-standing problem. More specifically, we show that the Beurling-type theorem holds in the Bergman space $A^2_α(D)$ for any $-1<α< +\infty$. That is, every invariant subspace $H$ for the shift operator $S$ on $A^2_α(D)$ $(-1<α< +\infty)$ has the property $H=[H\ominus zH]_{S,A^2_α\left(D\right)}$.
The results in the paper are related to the classification problem for invariant subspaces of multiplication operators in several variables. The main results consist of characterizations, in the two dimensional case, of ideals of polynomials and analytic functions which are closed in the relative topology induced by Bergman spaces on certain domains. This provides proofs of the Bergman space analogues of a conjecture of R. G. Douglas and V. Paulsen.
Suppose that $f := (f_1,\ldots,f_d):Ω_1\toΩ_2$ is a proper holomorphic map between two bounded domains in $\mathbb C^d.$ In this paper, we find a non-trivial minimal joint reducing subspace for the multiplication operator (tuple) $M_f=(M_{f_1},\ldots, M_{f_d})$ on the Bergman space $\mathbb A^2(Ω_1)$, say $\mathcal M.$ We further show that the restriction of $(M_{f_1},\ldots,M_{f_d})$ to $\mathcal M$ is unitarily equivalent to Bergman operator on $\mathbb A^2(Ω_2).$
It is known that (i) a subspace ${\mathcal N}$ of the Hardy space $H^2$ which is invariant under the backward shift operator can be represented as the range of the observability operator of a conservative discrete-time linear system, (ii) the transfer-function of this conservative linear system in turn is the inner Beurling-Lax representer for the forward-shift invariant subspace ${\mathcal M} : = {\mathcal N}^\perp$, and (iii) this transfer function also serves as the Sz.-Nagy-Foias characteristic function of the pure contraction operator $T$ given by $T = P_{\mathcal N} M_z |_{\mathcal N}$. …
In this paper, using the group-like property of local inverses of a finite Blaschke product $ϕ$, we will show that the largest $C^*$-algebra in the commutant of the multiplication operator $M_ϕ$ by $ϕ$ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of $ϕ^{-1}\circϕ$ over the unit disk. If the order of the Blaschke product $ϕ$ is less than or equal to eight, then every $C^*$-algebra contained in the commutant of $M_ϕ$ is abelian and hence the number of minimal reducing subspaces of $M_ϕ$ equals the number of connect…
We study the invariant subspaces generated by inner functions for a class of $\mathcal{P}^t(μ)$-spaces which can be identified as spaces of analytic functions in the unit disk $\mathbb{D}$, where $μ$ is a measure supported in the closed unit disk and $\mathcal{P}^t(μ)$ is the span of analytic polynomials in the usual Lebesgue space $L^t(μ)$. Our measures define a range of spaces somewhere in between the Hardy and the Bergman spaces, and our results are thus a mixture of results from these two theories. For a large class of measures $μ$ we characterize the cyclic inner functions, and exhibit so…
We study reducing subspaces for an analytic multiplication operator M_{z^{n}} on the Bergman space L_{a}^{2}(A_{r}) of the annulus A_{r}, and we prove that M_{z^{n}} has exactly 2^n reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context of complex geometry.
Let $v(r)=\exp\left(-\fracα{1-r}\right)$ with $α>0$, and let $\mathbb{D}$ be the unit disc in the complex plane. Denote by $A^p_v$ the subspace of analytic functions of $L^p(\mathbb{D},v)$ and let $P_v$ be the orthogonal projection from $L^2(\mathbb{D},v)$ onto $A^2_v$. In 2004, Dostanic revealed the intriguing fact that $P_v$ is bounded from $L^p(\mathbb{D},v)$ to $A^p_v$ only for $p=2$, and he posed the related problem of identifying the duals of $A^p_v$ for $p\ge 1$, $p\neq 2$. In this paper we propose a solution to this problem by proving that $P_v$ is bounded from $\,L^p(\D,v^{p/2})$ to $…
Let T be a C_{\cdot 0}-contraction on a Hilbert space H and S be a non-trivial closed subspace of H. We prove that S is a T-invariant subspace of H if and only if there exists a Hilbert space D and a partially isometric operator Π: H^2_D(\mathbb{D}) \raro H such that ΠM_z = T Πand that S = ran Π, or equivalently, P_S = ΠΠ^*. As an application we completely classify the shift-invariant subspaces of C_{\cdot 0}-contractive and analytic reproducing kernel Hilbert spaces over the unit disc. Our results also includes the case of weighted Bergman spaces over the unit disk.
It is shown that the algebra \(L^\infty(μ)\) of all bounded measurable functions with respect to a finite measure \(μ\) is localizing on the Hilbert space \(L^2(μ)\) if and only if the measure \(μ\) has an atom. Next, it is shown that the algebra \(H^\infty({\mathbb D})\) of all bounded analytic multipliers on the unit disc fails to be localizing, both on the Bergman space \(A^2({\mathbb D})\) and on the Hardy space \(H^2({\mathbb D}).\) Then, several conditions are provided for the algebra generated by a diagonal operator on a Hilbert space to be localizing. Finally, a theorem is provided abo…
This paper is a follow-up contribution to our work [20] where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of [20] to the context of n-tuples of bounded linear operators on Hilbert spaces. Let T = (T_1, \ldots, T_n) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space \mathcal{H} and \mathcal{S} be a non-trivial closed subspace of \mathcal{H}. One of our main results states that: \mathcal{S} is a joint T-invariant subspace if and only if there exists a partially isometric operator Π\in \mathcal{B}(…
Interpretations of the Beurling-Lax-Halmos Theorem on invariant subspaces of the unilateral shift are explored using the language of Hilbert modules. Extensions and consequences are considered in both the one and multivariate cases with an emphasis on the classical Hardy, Bergman and Drury-Arveson spaces.
The weighted Bergman spaces on the polydisc, $\mathbb A^{(λ)}(\mathbb D^n)$, $λ>0,$ splits into orthogonal direct sum of subspaces $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ indexed by the partitions $\boldsymbol p$ of $n,$ which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on $n$ symbols. In this paper, we prove that each sub-module $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal to square of the dimension $χ_{\boldsymbol p}(1)$ of …
We study the strong continuity of weighted composition semigroups of the form $T_tf=φ_t'\left(f\circφ_t\right)$ in several spaces of analytic functions. First we give a general result on separable spaces and use it to prove that these semigroups are always strongly continuous in the Hardy and Bergman spaces. Then we focus on two non-separable family of spaces, the mixed norm and the weighted Banach spaces. We characterize the maximal subspace in which a semigroup of analytic functions induces a strongly continuous semigroup of weighted composition operators depending on its Denjoy-Wolff point,…
Let $μ_α$ be the Lebesgue plane measure on the unit disk with the radial weight $\frac{α+1}π(1-|z|^2)^α$. Denote by $\mathcal{A}^{2}_{n}$ the space of the $n$-analytic functions on the unit disk, square-integrable with respect to $μ_α$. Extending the results of Ramazanov (1999, 2002), we explain that disk polynomials (studied by Koornwinder in 1975 and Wünsche in 2005) form an orthonormal basis of $\mathcal{A}^{2}_{n}$. Using this basis, we provide the Fourier decomposition of $\mathcal{A}^{2}_{n}$ into the orthogonal sum of the subspaces associated with different frequencies. This leads to th…
The use of unitary invariant subspaces of a Hilbert space $\mathcal{H}$ is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of $L^2(\mathbb{R})$ and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows to merge the cases of finitely/infinitely generated unitary invariant subspaces former…
We study Toeplitz-type operators with respect to specific wavelets whose Fourier transforms are related to Laguerre polynomials. On the one hand, this choice of wavelets underlines the fact that these operators acting on wavelet subspaces share many properties with the classical Toeplitz operators acting on the Bergman spaces. On the other hand, it enables to study poly-Bergman spaces and Toeplitz operators acting on them from a different perspective. Restricting to symbols depending only on vertical variable in the upper half-plane of the complex plane these operators are unitarily equivalent…
Given an inner function $Θ$ in the unit disc $\mathbb{D}$, we study the boundedness of the differentiation operator which acts from from the model subspace $K\_Θ=\left(ΘH^{2}\right)^{\perp}$ of the Hardy space $H^{2},$ equiped with the $BMOA$-norm, to some radial-weighted Bergman space. As an application, we generalize Peller's inequality for Besov norms of rational functions $f$ of degree $n\geq1$ having no poles in the closed unit disc $\overline{\mathbb{D}}$.
We construct for each $0<p\le 1$ an infinite collection of subspaces of $\ell_p$ that extend the example from [J. Lindenstrauss, On a certain subspace of $\ell_{1}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 539-542] of a subspace of $\ell_{1}$ with no unconditional basis. The structure of this new class of $p$-Banach spaces is analyzed and some applications to the general theory of $\mathcal{L}_{p}$-spaces for $0<p<1$ are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in $p$-Banach spaces for $p<1$…
We answer the question of W.T. Gowers, giving an example of a bounded operator on a subspace of Gowers unconditional space which is not a strictly singular perturbation of a restriction of a diagonal operator. We make some observations on operators in arbitrary tight by support Banach space, showing in particular that in such space no two isomorphic infinitely dimensional subspaces form a direct sum.
A new subspace of Morrey spaces whose elements can be approximated by infinitely differentiable compactly supported functions is introduced. Consequently, we give an explicit description of the closure of the set of such functions in Morrey spaces. A generalization of known embeddings of Morrey spaces into weighted Lesbesgue spaces is also obtained.
In finitely-dimensional spaces the sum range of a series has to be an affine subspace. It is long known this is not the case in infinitely dimensional Banach spaces. In particular in 1984 M.I. Kadets and K. Wozniakowski obtained an example of a series the sum range of which consisted of two points, and asked whether it is possible to obtain more than two, but finitely many points. This paper answers the question positively, by showing how to obtain an arbitrary finite set as the sum range of a series in any infinitely dimensional Banach space.
Let $b(z) = \prod_{n=1}^\infty \frac{-\barα_n}{|α_n|} \frac{z - α_n}{1 - \barα_n z}$, where $\sum_{n=1}^\infty (1 - |α_n|) <\infty$, be the Blaschke product with zeros at $α_n \in \mathbb{D} \setminus \{0\}$. Then $\cls = \vee_{n=1}^\infty \big(z^n H^2(\mathbb{D})\big) \otimes \big(\prod_{k=n}^\infty \frac{-\barα_n}{|α_n|} \frac{z - α_n}{1 - \barα_n z} H^2(\mathbb{D})\big)$ is a joint $(M_{z_1}, M_{z_2})$ invariant subspace of the Hardy space $H^2(\mathbb{D}^2) \cong H^2(\mathbb{D}) \otimes H^2(\mathbb{D})$. This class of subspaces was originally introduced by Rudin in the context of infinite …
The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former space is equal to the latter one. This stronger property is necessary for some applications in the theory of valuations on manifolds.
Two methods of constructing infinitely many isomorphically distinct $\Cal L_p$-spaces have been published. In this article we show that these constructions yield very different spaces and in the process develop methods for dealing with these spaces from the isomorphic viewpoint. We use these methods to give a complete isomorphic classification of the spaces $R_p^α$ constructed by Bourgain, Rosenthal, and Schechtman and to show that $X_p\otimes X_p$ is not isomorphic to a complemented subspace of any $R_p^α.$
In this paper we initiate the study of a fundamental yet untapped random model of non-selfadjoint, bounded linear operators acting on a separable complex Hilbert space. We replace the weights $w_n=1$ in the classical unilateral shift $T$, defined as $Te_n=w_ne_{n+1}$, where $\{e_n\}_{n=1}^\infty$ form an orthonormal basis of a complex Hilbert space, by a sequence of i.i.d. random variables $\{X_n\}_{n=1}^{\infty}$; that is, $w_n=X_n$. This paper answers basic questions concerning such a model. We propose that this model can be studied in comparison with the classical Hardy/Bergman/Dirichlet sp…
