## Displaying similar documents to “Operator positivity and analytic models of commuting tuples of operators”

### Weighted sub-Bergman Hilbert spaces

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces ${A}_{\alpha }^{2}$, $-1<\alpha <\infty$. These spaces have already been studied in , ,  and . We extend some results from these papers.

### Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator ${}_{a,b}$ by ${}_{a,b}\left(f\right)\left(z\right)=\Gamma \left(a+1\right)/\Gamma \left(b+1\right){\int }_{0}^{1}\left(f\left(t\right){\left(1-t\right)}^{b}\right)/\left({\left(1-tz\right)}^{a+1}\right)dt$, where a and b are non-negative real numbers. In particular, for a = b = β, ${}_{a,b}$ becomes the generalized Hilbert operator ${}_{\beta }$, and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that ${}_{a,b}$ is bounded on Dirichlet-type spaces ${S}^{p}$, 0 < p < 2, and on Bergman spaces ${A}^{p}$, 2 < p < ∞. Also we find an upper bound for the norm of the operator ${}_{a,b}$....

### Order bounded composition operators on the Hardy spaces and the Nevanlinna class

Studia Mathematica

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We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces ${H}^{p}$ 0 < p < ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,Lh-order bounded (we write (X,Lh)-ob) with $X={H}^{p}$ or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into ${L}_{h}$. We give...

### Operators on a Hilbert space similar to a part of the backward shift of multiplicity one

Studia Mathematica

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Let A: X → X be a bounded operator on a separable complex Hilbert space X with an inner product $⟨·,·{⟩}_{X}$. For b, c ∈ X, a weak resolvent of A is the complex function of the form $⟨{\left(I-zA\right)}^{-1}b,c{⟩}_{X}$. We will discuss an equivalent condition, in terms of weak resolvents, for A to be similar to a restriction of the backward shift of multiplicity 1.

### Toeplitz operators on Bergman spaces and Hardy multipliers

Studia Mathematica

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We study Toeplitz operators ${T}_{a}$ with radial symbols in weighted Bergman spaces ${A}_{\mu }^{p}$, 1 < p < ∞, on the disc. Using a decomposition of ${A}_{\mu }^{p}$ into finite-dimensional subspaces the operator ${T}_{a}$ can be considered as a coefficient multiplier. This leads to new results on boundedness of ${T}_{a}$ and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of ${T}_{a}$ for a satisfying an assumption on the positivity of certain...

### On the separation properties of ${K}_{\omega }$

Colloquium Mathematicae

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### A classification of projectors

Banach Center Publications

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A positive operator A and a closed subspace of a Hilbert space ℋ are called compatible if there exists a projector Q onto such that AQ = Q*A. Compatibility is shown to depend on the existence of certain decompositions of ℋ and the ranges of A and ${A}^{1/2}$. It also depends on a certain angle between A() and the orthogonal of .

### Bounded evaluation operators from ${H}^{p}$ into ${\ell }^{q}$

Studia Mathematica

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Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by ${T}_{z,p}\left(f\right)={\left(1-|zₙ|²\right)}^{1/p}f\left(zₙ\right)$. Necessary and sufficient conditions on zₙ are given such that ${T}_{z,p}$ maps the Hardy space ${H}^{p}$ boundedly into the sequence space ${\ell }^{q}$. A corresponding result for Bergman spaces is also stated.

### Extension operators on balls and on spaces of finite sets

Studia Mathematica

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We study extension operators between spaces of continuous functions on the spaces $\sigma ₙ\left({2}^{X}\right)$ of subsets of X of cardinality at most n. As an application, we show that if ${B}_{H}$ is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator $T:C\left(\lambda {B}_{H}\right)\to C\left(\mu {B}_{H}\right)$.

### A Hankel matrix acting on Hardy and Bergman spaces

Studia Mathematica

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Let μ be a finite positive Borel measure on [0,1). Let ${ℋ}_{\mu }={\left({\mu }_{n,k}\right)}_{n,k\ge 0}$ be the Hankel matrix with entries ${\mu }_{n,k}={\int }_{\left[0,1\right)}{t}^{n+k}d\mu \left(t\right)$. The matrix ${}_{\mu }$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula ${ℋ}_{\mu }\left(f\right)\left(z\right)={\sum }_{n=0}^{\infty }i\left({\sum }_{k=0}^{\infty }{\mu }_{n,k}{a}_{k}\right)zⁿ$, z ∈ , where $f\left(z\right)={\sum }_{n=0}^{\infty }aₙzⁿ$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that ${ℋ}_{\mu }\left(f\right)\left(z\right)={\int }_{\left[0,1\right)}f\left(t\right)/\left(1-tz\right)d\mu \left(t\right)$ for all f in the Hardy space H¹, and among them we describe those for which ${ℋ}_{\mu }$ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A². ...

### Moore-Penrose inverses of Gram operators on Hilbert C*-modules

Studia Mathematica

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Let t be a regular operator between Hilbert C*-modules and ${t}^{†}$ be its Moore-Penrose inverse. We investigate the Moore-Penrose invertibility of the Gram operator t*t. More precisely, we study some conditions ensuring that ${t}^{†}={\left(t*t\right)}^{†}t*=t*{\left(tt*\right)}^{†}$ and ${\left(t*t\right)}^{†}={t}^{†}t{*}^{†}$. As an application, we get some results for densely defined closed operators on Hilbert C*-modules over C*-algebras of compact operators.

### Strict plurisubharmonicity of Bergman kernels on generalized annuli

Annales Polonici Mathematici

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Let ${A}_{\zeta }=\Omega -\overline{\rho \left(\zeta \right)·\Omega }$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel ${K}_{\zeta }\left(z\right)$ of ${A}_{\zeta }$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that ${A}_{\zeta }$ is non-pseudoconvex when the dimension of ${A}_{\zeta }$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ²log{K}_{\zeta }/\partial \zeta \partial \zeta ̅$, as well as its boundary behavior.

### Cyclic phenomena for composition operators on weighted Bergman spaces

Bollettino dell'Unione Matematica Italiana

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In the present paper we give a generalization to the family of Bergman Spaces with weight $G$, $A^{2}_{G}$ of several results, obtained in  for the Hardy space $H^{2}$, concerning the cyclic and hypercyclic behaviour of composition operators CW induced by a holomorphic self map $\varphi$ of the open unit disc $\Delta\subset\mathbb{C}$.

### On the range-kernel orthogonality of elementary operators

Mathematica Bohemica

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Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $\left(A,B\right)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...

### On isometrical extension properties of function spaces

Commentationes Mathematicae Universitatis Carolinae

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In this note, we prove that any “bounded” isometries of separable metric spaces can be represented as restrictions of linear isometries of function spaces $C\left(Q\right)$ and $C\left(\Delta \right)$, where $Q$ and $\Delta$ denote the Hilbert cube ${\left[0,1\right]}^{\infty }$ and a Cantor set, respectively.

### Essential norms of weighted composition operators between Hardy spaces ${H}^{p}$ and ${H}^{q}$ for 1 ≤ p,q ≤ ∞

Studia Mathematica

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We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces ${H}^{p}$ and ${H}^{q}$ for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.

### Non supercyclic subsets of linear isometries on Banach spaces of analytic functions

Czechoslovak Mathematical Journal

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Let $X$ be a Banach space of analytic functions on the open unit disk and $\Gamma$ a subset of linear isometries on $X$. Sufficient conditions are given for non-supercyclicity of $\Gamma$. In particular, we show that the semigroup of linear isometries on the spaces ${S}^{p}$ ($p>1$), the little Bloch space, and the group of surjective linear isometries on the big Bloch space are not supercyclic. Also, we observe that the groups of all surjective linear isometries on the Hardy space ${H}^{p}$ or the Bergman space ${L}_{a}^{p}$ ($1, $p\ne 2$)...

### New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces

Czechoslovak Mathematical Journal

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Let $u$ be a holomorphic function and $\varphi$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u{C}_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$, $\varphi$, their derivatives, and ${\varphi }^{n}$, the $n$-th power of $\varphi$. Moreover, we obtain some similar estimates for the essential norms of the operators $u{C}_{\varphi }$, from which sufficient and necessary conditions of compactness of $u{C}_{\varphi }$ follows immediately. ...

### Weighted ${H}^{p}$ spaces

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CONTENTSIntroduction.......................................................................................................................................................... 5Chapter I. Some preliminary lemmas............................................................................................................ 8Chapter II. Weighted ${H}^{p}$ spaces of analytic functions.......................................................................... 13 1. Behaviour at the boundary..........................................................................................................................