Operator positivity and analytic models of commuting tuples of operators

Monojit Bhattacharjee; Jaydeb Sarkar

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

Weighted sub-Bergman Hilbert spaces

Maria Nowak, Renata Rososzczuk (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

We consider Hilbert spaces which are counterparts of the de Branges-Rovnyak spaces in the context of the weighted Bergman spaces $A_{α}^{2}$ , $- 1 < α < \infty$ . These spaces have already been studied in [8], [7], [5] and [1]. We extend some results from these papers.

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Sunanda Naik, Karabi Rajbangshi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator $_{a, b}$ by $_{a, b} (f) (z) = Γ (a + 1) / Γ (b + 1) \int_{0}^{1} (f (t) {(1 - t)}^{b}) / ({(1 - t z)}^{a + 1}) d t$ , where a and b are non-negative real numbers. In particular, for a = b = β, $_{a, b}$ becomes the generalized Hilbert operator $_{β}$ , 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 $_{a, b}$ ....

Numerical radius inequalities for Hilbert $C^{*}$ -modules

Sadaf Fakri Moghaddam, Alireza Kamel Mirmostafaee (2022)

Mathematica Bohemica

Similarity:

We present a new method for studying the numerical radius of bounded operators on Hilbert $C^{*}$ -modules. Our method enables us to obtain some new results and generalize some known theorems for bounded operators on Hilbert spaces to bounded adjointable operators on Hilbert $C^{*}$ -module spaces.

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

Nizar Jaoua (1999)

Studia Mathematica

Similarity:

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...

Some equivalent metrics for bounded normal operators

Mohammad Reza Jabbarzadeh, Rana Hajipouri (2018)

Mathematica Bohemica

Similarity:

Some stronger and equivalent metrics are defined on $ℳ$ , the set of all bounded normal operators on a Hilbert space $H$ and then some topological properties of $ℳ$ are investigated.

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

Yoichi Uetake (2001)

Studia Mathematica

Similarity:

Let A: X → X be a bounded operator on a separable complex Hilbert space X with an inner product $⟨ \cdot, \cdot ⟩_{X}$ . For b, c ∈ X, a weak resolvent of A is the complex function of the form $⟨ {(I - z A)}^{- 1} b, c ⟩_{X}$ . 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.

Hilbert-Schmidt Hankel operators with anti-holomorphic symbols on a class of unbounded complete Reinhardt domains

Le He, Yanyan Tang (2024)

Czechoslovak Mathematical Journal

Similarity:

We consider a class of unbounded nonhyperbolic complete Reinhardt domains $D_{n, m, k}^{μ, p, s} : = \{(z, w_{1}, \dots, w_{m}) \in ℂ^{n} \times ℂ^{k_{1}} \times \dots \times ℂ^{k_{m}} : \frac{∥ w_{1} ∥^{2 p_{1}}}{e^{- μ_{1} {∥ z ∥}^{s}}} + \dots + \frac{∥ w_{m} ∥^{2 p_{m}}}{e^{- μ_{m} {∥ z ∥}^{s}}} < 1\},$ where $s$ , $p_{1}, \dots, p_{m}$ , $μ_{1}, \dots, μ_{m}$ are positive real numbers and $n$ , $k_{1}, \dots, k_{m}$ are positive integers. We show that if a Hankel operator with anti-holomorphic symbol is Hilbert-Schmidt on the Bergman space $A^{2} (D_{n, m, k}^{μ, p, s})$ , then it must be zero. This gives an example of high dimensional unbounded complete Reinhardt domain that does not admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.

Toeplitz operators on Bergman spaces and Hardy multipliers

Wolfgang Lusky, Jari Taskinen (2011)

Studia Mathematica

Similarity:

We study Toeplitz operators $T_{a}$ with radial symbols in weighted Bergman spaces $A_{μ}^{p}$ , 1 < p < ∞, on the disc. Using a decomposition of $A_{μ}^{p}$ 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 $K_{ω}$

Malgorzata Wójcicka (1986)

Colloquium Mathematicae

Similarity:

A classification of projectors

Gustavo Corach, Alejandra Maestripieri, Demetrio Stojanoff (2005)

Banach Center Publications

Similarity:

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 $ℓ^{q}$

Martin Smith (2007)

Studia Mathematica

Similarity:

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

Extension operators on balls and on spaces of finite sets

Antonio Avilés, Witold Marciszewski (2015)

Studia Mathematica

Similarity:

We study extension operators between spaces of continuous functions on the spaces $σ ₙ (2^{X})$ 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 $T : C (λ B_{H}) \to C (μ B_{H})$ .

A Hankel matrix acting on Hardy and Bergman spaces

Petros Galanopoulos, José Ángel Peláez (2010)

Studia Mathematica

Similarity:

Let μ be a finite positive Borel measure on [0,1). Let $ℋ_{μ} = {(μ_{n, k})}_{n, k \geq 0}$ be the Hankel matrix with entries $μ_{n, k} = \int_{[0, 1)} t^{n + k} d μ (t)$ . The matrix $_{μ}$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula $ℋ_{μ} (f) (z) = \sum_{n = 0}^{\infty} i (\sum_{k = 0}^{\infty} μ_{n, k} a_{k}) z ⁿ$ , z ∈ , where $f (z) = \sum_{n = 0}^{\infty} a ₙ z ⁿ$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that $ℋ_{μ} (f) (z) = \int_{[0, 1)} f (t) / (1 - t z) d μ (t)$ for all f in the Hardy space H¹, and among them we describe those for which $ℋ_{μ}$ 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

M. S. Moslehian, K. Sharif, M. Forough, M. Chakoshi (2012)

Studia Mathematica

Similarity:

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 $t^{†} = {(t * t)}^{†} t * = t * {(t t *)}^{†}$ and ${(t * t)}^{†} = t^{†} t *^{†}$ . As an application, we get some results for densely defined closed operators on Hilbert C*-modules over C*-algebras of compact operators.

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Suying Liu, Minghua Yang (2018)

Czechoslovak Mathematical Journal

Similarity:

Let $L$ be a non-negative self-adjoint operator acting on $L^{2} (ℝ^{n})$ satisfying a pointwise Gaussian estimate for its heat kernel. Let $w$ be an $A_{r}$ weight on $ℝ^{n} \times ℝ^{n}$ , $1 < r < \infty$ . In this article we obtain a weighted atomic decomposition for the weighted Hardy space $H_{L, w}^{p} (ℝ^{n} \times ℝ^{n})$ , $0 < p \leq 1$ associated to $L$ . Based on the atomic decomposition, we show the dual relationship between $H_{L, w}^{1} (ℝ^{n} \times ℝ^{n})$ and ${BMO}_{L, w} (ℝ^{n} \times ℝ^{n})$ .

Generalized atomic subspaces for operators in Hilbert spaces

Prasenjit Ghosh, Tapas Kumar Samanta (2022)

Mathematica Bohemica

Similarity:

We introduce the notion of a $g$ -atomic subspace for a bounded linear operator and construct several useful resolutions of the identity operator on a Hilbert space using the theory of $g$ -fusion frames. Also, we shall describe the concept of frame operator for a pair of $g$ -fusion Bessel sequences and some of their properties.

Strict plurisubharmonicity of Bergman kernels on generalized annuli

Yanyan Wang (2014)

Annales Polonici Mathematici

Similarity:

Let $A_{ζ} = Ω - \bar{ρ (ζ) \cdot Ω}$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel $K_{ζ} (z)$ of $A_{ζ}$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that $A_{ζ}$ is non-pseudoconvex when the dimension of $A_{ζ}$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ² l o g K_{ζ} / \partial ζ \partial ζ ̅$ , as well as its boundary behavior.

Hilbert series of the Grassmannian and $k$ -Narayana numbers

Lukas Braun (2019)

Communications in Mathematics

Similarity:

We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the $q$ -Hilbert series is a Vandermonde-like determinant. We show that the $h$ -polynomial of the Grassmannian coincides with the $k$ -Narayana polynomial. A simplified formula for the $h$ -polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the $k$ -Narayana numbers,...

Cyclic phenomena for composition operators on weighted Bergman spaces

Anna Gori (2006)

Bollettino dell'Unione Matematica Italiana

Similarity:

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 [4] 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

Said Bouali, Youssef Bouhafsi (2015)

Mathematica Bohemica

Similarity:

Let $L (H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$ . For $A, B \in L (H)$ , the generalized derivation $δ_{A, B}$ and the elementary operator $Δ_{A, B}$ are defined by $δ_{A, B} (X) = A X - X B$ and $Δ_{A, B} (X) = A X B - X$ for all $X \in L (H)$ . In this paper, we exhibit pairs $(A, B)$ of operators such that the range-kernel orthogonality of $δ_{A, B}$ 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 $Δ_{A, B}$ with respect to the wider class of unitarily invariant...