scholarly journals Toeplitz Operators on the Symmetrized Bidisc

2020 ◽  
Author(s):  
Tirthankar Bhattacharyya ◽  
B Krishna Das ◽  
Haripada Sau
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Unit Disc ◽  
Toeplitz Operators ◽  
Reproducing Kernel ◽  
Multiplication Operators ◽  
Algebraic Characterization ◽  
Hilbert Modules ◽  
Symmetrized Bidisc

Abstract The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

scholarly journals Power series of several variables with condition of logarithmical convexity

2021 ◽  
Vol 8 (1) ◽  
pp. 49-62
Author(s):  
Alexandr V. Zheleznyak ◽  
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Power Series ◽  
Unit Disc ◽  
Reproducing Kernel ◽  
Space Of Analytic Functions ◽  
Several Variables ◽  
Positive Coefficients ◽  
Classical Hardy Space ◽  
Reproducing Kernel Function

We obtain a new version of Hardy theorem about power series of several variables reciprocal to the power series with positive coefficients. We prove that if the sequence {as} = as1,s2,...,sn, ||s|| ≥ K satisfies condition of logarithmically convexity and the first coefficient a0 is sufficiently large then reciprocal power series has only negative coefficients {bs} = bs1,s2,...,sn, except b0,0,...,0 for any K. The classical Hardy theorem corresponds to the case K = 0, n = 1. Such results are useful in Nevanlinna - Pick theory. For example, if function k(x, y) can be represented as power series Σn≥0 an(x-y)n, an > 0, and reciprocal function 1/k(x,y) can be represented as power series Σn≥0 bn(x-y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna-Pick property. The reproducing kernel 1/1-x-y of the classical Hardy space H2(D) is a prime example for our theorems.

Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

2021 ◽  
Vol 56 (2) ◽  
pp. 343-374
Author(s):  
Boris Guljaš ◽  
Keyword(s):  
Hilbert Space ◽  
Hilbert Modules ◽  
Strict Topology ◽  
Direct Sums ◽  
Essential Ideal ◽  
Closed Submodule ◽  
Closed Submodules ◽  
Hereditary Algebras ◽  
Multiplier Algebras

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

Online regularized pairwise learning with least squares loss

2019 ◽  
Vol 18 (01) ◽  
pp. 49-78 ◽  
Author(s):  
Cheng Wang ◽  
Ting Hu
Keyword(s):  
Hilbert Space ◽  
Least Squares ◽  
Online Algorithm ◽  
Reproducing Kernel ◽  
Sharp Estimate ◽  
Reproducing Kernel Hilbert Space ◽  
Learning Sequence ◽  
Pairwise Learning ◽  
Regularization Parameters

In this paper, we study online algorithm for pairwise problems generated from the Tikhonov regularization scheme associated with the least squares loss function and a reproducing kernel Hilbert space (RKHS). This work establishes the convergence for the last iterate of the online pairwise algorithm with the polynomially decaying step sizes and varying regularization parameters. We show that the obtained error rate in [Formula: see text]-norm can be nearly optimal in the minimax sense under some mild conditions. Our analysis is achieved by a sharp estimate for the norms of the learning sequence and the characterization of RKHS using its associated integral operators and probability inequalities for random variables with values in a Hilbert space.

The reproducing kernel of $\mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian

2019 ◽  
Vol 124 (1) ◽  
pp. 81-101
Author(s):  
Manfred Stoll
Keyword(s):  
Hardy Space ◽  
Unit Ball ◽  
Hypergeometric Function ◽  
Compact Subset ◽  
Spherical Harmonics ◽  
Harmonic Functions ◽  
Reproducing Kernel ◽  
Diagonal Function ◽  
Hyperbolic Harmonic Functions

In the paper we characterize the reproducing kernel $\mathcal {K}_{n,h}$ for the Hardy space $\mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $\mathbb {B}$ in $\mathbb {R}^n$. Specifically we prove that \[ \mathcal {K}_{n,h}(x,y) = \sum _{\alpha =0}^\infty S_{n,\alpha }(\lvert x\rvert )S_{n,\alpha }(\lvert y\rvert ) Z_\alpha (x,y), \] where the series converges absolutely and uniformly on $K\times \mathbb {B}$ for every compact subset $K$ of $\mathbb {B}$. In the above, $S_{n,\alpha }$ is a hypergeometric function and $Z_\alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0\le \mathcal K_{n,h}(x,y) \le \frac {C_n}{(1-2\langle x,y\rangle + \lvert x \rvert^2 \lvert y \rvert^2)^{n-1}}, \] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $\mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $\varDelta_h $ on $\mathbb{B} $ with eigenvalue $\lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $\varDelta_h $ on $\mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $\varDelta_h $ with eigenvalue $\lambda _\alpha = 4(n-1)^2\alpha (\alpha -1)$, then \[ g(r) = g(0) \frac {p_{n,\alpha }(r^2)}{(1-r^2)^{(\alpha -1)(n-1)}}, \] where $p_{n,\alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,\alpha }(r^2)$ is a polynomial of degree $2(\alpha -1)(n-1)$.

scholarly journals Bilinear forms on vector Hardy spaces

1997 ◽  
Vol 39 (3) ◽  
pp. 371-378
Author(s):  
Gordon Blower
Keyword(s):  
Hilbert Space ◽  
Hardy Space ◽  
Bilinear Form ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Reproducing Kernel ◽  
Bilinear Forms ◽  
Sufficient Condition

AbstractLet φ: ℋ → be a bilinear form on vector Hardy space. Introduce the symbol φ of Φ by (φ (Z1, Z2), a ⊗ b) = Φ (K21 ⊗ a, K22 ⊗ b ), where Kw is the reproducing kernel for w ∈ D. We show that Φ extends to a bounded bilinear form on provided that the gradient defines a Carleson measure in the bidisc D2. We obtain a sufficient condition for Φ to extend to a Hilbert space. For vectorial bilinear Hankel forms we obtain an analogue of Nehari's Theorem.

scholarly journals Lebesgue Decomposition of Non-Commutative Measures

2020 ◽  
Author(s):  
Michael T Jury ◽  
Robert T W Martin
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Reproducing Kernel Hilbert Space ◽  
Semigroup Algebra ◽  
Space Theory ◽  
Lebesgue Decomposition ◽  
Sesquilinear Forms ◽  
Positive Linear Functionals ◽  
Algebra Theory

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

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