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Composition operators with analytic symbols on some reproducing kernel Hilbert spaces of entire functions on a complex Hilbert space are studied. The questions of their boundedness, seminormality and positivity are investigated. It is proved that if such an operator is bounded, then its symbol is a polynomial of degree at most 1, i.e., it is an affine mapping. Fock's type model for composition operators with linear symbols is established. As a consequence, explicit formulas for their polar decomposition, Aluthge transform and powers with positive real exponents are provided. The theorem of Carswell, MacCluer and Schuster is generalized to the case of Segal-Bargmann spaces of infinite order. Some related questions are also discussed.

The Cesaro operator $\mathsf{C}$, when acting in the classical growth Banach spaces $A^{-\gamma}$ and $A_0^{-\gamma}$, for $\gamma > 0 $, of analytic functions on $\mathbb{D}$, is investigated. Based on a detailed knowledge of their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the norms of these operators precisely. It is then possible to characterize the mean ergodic and related properties of $\mathsf{C}$ acting in these spaces. In addition, we determine the largest Banach space of analytic functions on $\mathbb{D}$ which $\mathsf{C}$ maps into $A^{-\gamma}$ (resp. into $A_0^{-\gamma}$); this optimal domain space always contains $A^{-\gamma}$ (resp. $A_0^{-\gamma}$) as a proper subspace.

We study weighted $H^\infty$ spaces of analytic functions on the open unit disc in the case of non-doubling weights, which decrease rapidly with respect to the boundary distance. We characterize the solid hulls of such spaces and give quite explicit representations of them in the case of the most natural exponentially decreasing weights.

Using the Cauchy-Riemann operator, we characterize $Q_K$ spaces, Besov spaces and analytic Morrey spaces in terms of pseudoanalytic extensions of primitive functions. Our results are also true on some classical Banach spaces, such as the Bloch space, $BMOA$ and the Dirichlet space.

The notion of a Carleson measure was introduced by Lennart Carleson in his proof of the Corona Theorem for $H^\infty(\mathbb{D})$. In this paper we will define it for certain type of reproducing kernel Hilbert spaces of analytic functions of the complex half-plane, $\mathbb{C}_+$, which will include Hardy, Bergman and Dirichlet spaces. We will obtain several necessary or sufficient conditions for a positive Borel measure to be Carleson by preforming tests on reproducing kernels, weighted Bergman kernels, and studying the tree model obtained from a decomposition of the complex half-plane. The Dirichlet space will be investigated in detail as a special case. Finally, we will present a control theory application of Carleson measures in determining admissibility of controls in well-posed linear evolution equations.

We characterize the compactness of composition operators; in term of generalized Nevanlinna counting functions, on a large class of Hilbert spaces of analytic functions, which can be viewed between the Bergman and the Dirichlet spaces

We characterize the compactness of composition operators; in term of generalized Nevanlinna counting functions, on a large class of Hilbert spaces of analytic functions, which can be viewed between the Bergman and the Dirichlet spaces

We prove some sharp extremal distance results for functions in weighted Bergman spaces on the upper halfplane.We also prove such results in the context of bounded strictly pseudoconvex domains with smooth boundary

We prove some sharp extremal distance results for functions in weighted Bergman spaces on the upper halfplane.We also prove such results in the context of bounded strictly pseudoconvex domains with smooth boundary

Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $\epsilon>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq \epsilon$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert space. Furthermore, in the Hilbertian case we also show that $C$ can be assumed to be any number greater than 1.

Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each {\epsilon}>0, there exists an analytic function $g:X \rightarrow R$ with $|g-f|

We obtain new characterizations of Bergman and Bloch spaces on the unit disc involving equivalent (quasi)-norms of these spaces.Our results are in spirit of estimates obtained by Fefferman and Stein for HArdy spaces in R^n.

We study integration in a class of Hilbert spaces of analytic functions defined on the $\mathbb{R}^s$. The functions are characterized by the property that their Hermite coefficients decay exponentially fast. We use Gauss-Hermite integration rules and show that the error of our algorithms decays exponentially fast. Furthermore, we give necessary and sufficient conditions under which we achieve exponential convergence with weak, polynomial, and strong polynomial tractability.

Let {\phi} be an analytic self-map of D and be an analytic operator-valued function on D, where D is the unit disk. We provide necessary and sufficient conditions for the boundedness and compactness of weighted composition operators W_{\psi,\phi} on Lip_A(D;X;\alpha) and lip_A(D;X;{\alpha}), the spaces of analytic X-valued Lipschitz functions f, where X is a complex Banach space and {\alpha} is in (0, 1].

In the paper compact multiplier operators on Banach spaces of analytic functions on the unit disk with the range in Banach sequence lattices are studied. If the domain space $X$ is such that $H_\infty\hookrightarrow X\hookrightarrow H_1$, necessary and sufficient conditions for compactness are presented. Moreover, the calculation of the Hausdorff measure of noncompactness for diagonal operators between Banach sequence lattices is applied to obtaining the characterization of compact multipliers in case the domain space $X$ satisfies $H_\infty\hookrightarrow X\hookrightarrow H_2$.

We consider weighted composition operators on spaces of analytic functions on the unit disc, which take values in some complex Banach space. We provide necessary and sufficient conditions for the boundedness and (weak) compactness of weighted composition operators on general function spaces, and in particular on weak vector-valued spaces. As an application, we characterize the weak compactness of these operators between two different vector-valued Bloch-type spaces. This result appears to be new also in the scalar-valued case.

We study the asymptotics near the origin of the Fourier transform in weighted Hardy spaces of analytic functions in the upper half-plane, and of the Laplace transform in weighted spaces of entire functions of zero exponential type. These results are applied to two closely related problems posed by E. Dyn'kin: we find the asymptotics of the depth of zero in non-quasianalytic Denjoy-Carleman classes, and of the exact Levinson-Sjoberg majorant.

In this paper the spectrum of composition operators on the space of real analytic functions is investigated. In some cases it is completely determined while in some other cases it is only estimated.

We study multivariate $L_2$-approximation for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences $\boldsymbol{a} =\{a_j\}$ and $\boldsymbol{b} =\{b_j\}$ of numbers no less than one. Let $e^{L_2-\mathrm{app},\Lambda}(n,s)$ be the minimal worst-case error of all algorithms that use $n$ information functionals from the class $\Lambda$ in the $s$-variate case. We consider two classes $\Lambda$: the class $\Lambda^{{\rm all}}$ consists of all linear functionals and the class $\Lambda^{{\rm std}}$ consists of only function valuations. We study (EXP) exponential convergence. This means that $$ e^{L_2-\mathrm{app},\Lambda}(n,s) \le C(s)\,q^{\,(n/C_1(s))^{p(s)}}\quad{for all}\quad n, s \in \mathbb{N} $$ where $q\in(0,1)$, and $C,C_1,p:\mathbb{N} \rightarrow (0,\infty)$. If we can take $p(s)=p>0$ for all $s$ then we speak of (UEXP) uniform exponential convergence. We also study EXP and UEXP with (WT) weak, (PT) polynomial and (SPT) strong polynomial tractability. These concepts are defined as follows. Let $n(\e,s)$ be the minimal $n$ for which $e^{L_2-\mathrm{app},\Lambda}(n,s)\le \e$. Then WT holds iff $\lim_{s+\log\,\e^{-1}\to\infty}(\log n(\e,s))/(s+\log\,\e^{-1})=0$, PT holds iff there are $c,\tau_1,\tau_2$ such that $n(\e,s)\le cs^{\tau_1}(1+\log\,\e^{-1})^{\tau_2}$ for all $s$ and $\e\in(0,1)$, and finally SPT holds iff the last estimate holds for $\tau_1=0$. The infimum of $\tau_2$ for which SPT holds is called the exponent $\tau^*$ of SPT. We prove that the results are the same for both classes $\Lambda$, and obtain conditions for WT, PT, SPT with and without EXP and UEXP.

A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map $f$ of the open unit disk there is even an indestructible Blaschke product $B$ such that the critical sets of $f$ and $B$ coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighted Bergman space as well as to the Berger-Nirenberg problem from differential geometry. By solving the Berger-Nirenberg problem for a special case we identify the critical sets of bounded analytic functions with the zero sets of the weighted Bergman space ${\cal A}_1^2$.

A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map $f$ of the open unit disk there is even an indestructible Blaschke product $B$ such that the critical sets of $f$ and $B$ coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighted Bergman space as well as to the Berger-Nirenberg problem from differential geometry. By solving the Berger-Nirenberg problem for a special case we identify the critical sets of bounded analytic functions with the zero sets of the weighted Bergman space ${\cal A}_1^2$.

In this work we study the essential spectra of composition operators on Hardy spaces of analytic functions which might be termed as "quasi-parabolic". This is the class of composition operators on H^{2} with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form \phi(z) = z+\psi(z) where \psi\in H^{2}(\mathbb{H}) and \Im(\psi(z)) >\delta > 0. We especially examine the case where \psi is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.

We obtain sampling and interpolation theorems in radial weighted spaces of analytic functions for weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted Bergman spaces.

We obtain sampling and interpolation theorems in radial weighted spaces of analytic functions for weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted Bergman spaces.

We obtain sampling and interpolation theorems in radial weighted spaces of analytic functions for weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted Bergman spaces.

It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted $L^2$ functions defined on $(0, \infty)$ and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic functions are not only Hilbert spaces but also Banach algebras, and are therefore contained within their spaces of multipliers. We provide some necessary as well as sufficient conditions for this to happen and look at its consequences.

We investigate tameness in the Fr\'echet spaces O(M) of analytic functions on Stein manifolds M equipped with the compact open topology. Actually we will look into tameness in the more general class of nuclear Fr\'echet spaces with the properties weak DN and Omega, and then specialize to analytic function spaces. We show that for a Stein manifold M, tameness of O(M) is equivalent to the hyperconvexity of M.

In this paper, the $m-$order infinite dimensional Hilbert tensor (hypermatrix) is intrduced to define an $(m-1)$-homogeneous operator on the spaces of analytic functions, which is called Hilbert tensor operator. The boundedness of Hilbert tensor operator is presented on Bergman spaces $A^p$ ($p>2(m-1)$). On the base of the boundedness, two positively homogeneous operators are introduced to the spaces of analytic functions, and hence the upper bounds of norm of such two operators are found on Bergman spaces $A^p$ ($p>2(m-1)$). In particular, the norms of such two operators on Bergman spaces $A^{4(m-1)}$ are smaller than or equal to $\pi$ and $\pi^\frac1{m-1}$, respectively.

We consider different characterizations of Triebel--Lizorkin type spaces of analytic functions on the unit disc. Even though our results appear in the folklore, detailed descriptions are hard to find, and in fact we are unable to discuss the full range of parameters. Without additional effort we work with vector-valued analytic functions, and also consider a generalized scale of function spaces, including for example so-called $Q$-spaces. The primary aim of this note is to generalize, and clarify, a remarkable result by Cohn and Verbitsky, on factorization of Triebel--Lizorkin spaces. Their result remains valid for functions taking values in an arbitrary Banach space, provided that the vector-valuedness "sits in the right factor". On the other hand, if we impose vector-valuedness on the "wrong" factor, then the factorization fails even for separable Hilbert spaces.

In this paper we will show how the boundedness condition for the weighted composition operators on a class of spaces of analytic functions on the open right complex half-plane called Zen spaces (which include the Hardy spaces and weighted Bergman spaces) can be stated in terms of Carleson measures and Bergman kernels. In Hilbertian setting we will also show how the norms of \emph{causal} weighted composition operators on these spaces are related to each other and use it to show that an \emph{(unweighted) composition operator} $C_\varphi$ is bounded on a Zen space if and only if $\varphi$ has a finite angular derivative at infinity. Finally, we will show that there is no compact composition operator on Zen spaces.

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of $\mathbb{C}.$ In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued $\mathrm{H}^1$-BMOA duality theorem is proved. Finally, by the $\mathrm{H}^1$-BMOA duality we present the Lusin area integral and Littlewood-Paley $g$-function characterizations of the operator-valued analytic Hardy space.

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of $\mathbb{C}.$ In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued $\mathrm{H}^1$-BMOA duality theorem is proved. Finally, by the $\mathrm{H}^1$-BMOA duality we present the Lusin area integral and Littlewood-Paley $g$-function characterizations of the operator-valued analytic Hardy space.

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of $\mathbb{C}.$ In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued $\mathrm{H}^1$-BMOA duality theorem is proved. Finally, by the $\mathrm{H}^1$-BMOA duality we present the Lusin area integral and Littlewood-Paley $g$-function characterizations of the operator-valued analytic Hardy space.

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of $\mathbb{C}.$ In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued $\mathrm{H}^1$-BMOA duality theorem is proved. Finally, by the $\mathrm{H}^1$-BMOA duality we present the Lusin area integral and Littlewood-Paley $g$-function characterizations of the operator-valued analytic Hardy space.

Let D be a strictly convex domain and X be a singular analytic subset of C^2 such that the intersection of X and D is non empty. We give conditions under which a function holomophic on the intersection of X and D can be extended holomorphically to D with growth estimates of BMO or L^q type. The extension is given by mean of integral representation formulas and new residual currents.

Let D be a strictly convex domain and X be a singular analytic subset of C^2 such that the intersection of X and D is non empty. We give conditions under which a function holomophic on the intersection of X and D can be extended holomorphically to D with growth estimates of BMO or L^q type. The extension is given by mean of integral representation formulas and new residual currents.

We study function spaces consisting of analytic functions with fast decay on horizontal strips of the complex plane with respect to a given weight function. Their duals, so called spaces of (ultra)hyperfunctions of fast growth, generalize the spaces of Fourier hyperfunctions and Fourier ultrahyperfunctions. An analytic representation theory for their duals is developed and applied to characterize the non-triviality of these function spaces in terms of the growth order of the weight function. In particular, we show that the Gelfand-Shilov spaces of Beurling type $\mathcal{S}^{(p!)}_{(M_p)}$ and Roumieu type $\mathcal{S}^{\{p!\}}_{\{M_p\}}$ are non-trivial if and only if $$ \sup_{p \geq 2}\frac{(\log p)^p}{h^pM_p} < \infty, $$ for all $h > 0$ and some $h > 0$, respectively. We also study boundary values of holomorphic functions in spaces of ultradistributions of exponential type, which may be of quasianalytic type.