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Shiv Nadar University

In this paper we extend the classical Bohr’s inequality to the setting of the non-commutative Hardy space H1 associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr’s inequality for operators in the von Neumann-Schatten class C1 and square matrices of any finite order. Interestingly, we establish that the optimal bound for r in the above mentioned Bohr’s inequality concerning von Neumann-Schatten class is 1/3 whereas it is 1/2 in the case of 2 × 2 matrices and reduces to √2 − 1 for the case of 3 × 3 matrices. We also obtain a generalization of our above-mentioned Bohr’s inequality for finite matrices where we show that the optimal bound for r, unlike above, remains 1/3 for every fixed order n × n, n ≥ 2. 2021 American Mathematical Society

Mathematics

(SoNS) School of Natural Sciences

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Proceedings of the American Mathematical Society

BOHR’S INEQUALITY for NON-COMMUTATIVE HARDY SPACES

This paper deals with representing in concrete fashion those Hilbert spaces that are vector subspaces of the Hardy spaces Hp(Dn)(1≤p≤∞) that remain invariant under the action of coordinate wise multiplication by an n-tuple (TB1,…,TBn) of operators where each Bi,1≤i≤n, is a finite Blaschke factor on the open unit disc. The critical point to be noted is that these TBi are assumed to be weaker than isometries as operators. Thus our main theorems extends the principal result of [10] in the following three directions: (i) from one to several variables; (ii) from multiplication with the coordinate function z to an n-tuple of multiplication by finite Blaschke factors Bi,1≤i≤n; (iii) from vector subspaces of H2(D) to the case of vector subspaces of Hp(Dn),1≤p≤∞. We further derive a generalization of Slocinski's well known Wold type decomposition of a pair of doubly commuting isometries to the case of n-tuple of doubly commuting operators whose actions are weaker than isometries. © 2022

Fulltext

Journal of Mathematical Analysis and Applications

Multivariable sub-Hardy Hilbert spaces invariant under the action of n-tuple of finite Blaschke factors

In this note we study sub-Hardy Hilbert spaces on which the the action of the operator of multiplication by the coordinate function z is assumed to be weaker than that of an isometry. We identify such operators with a class of weighted shifts. The well known results of de Branges and Beurling are deduced as corollaries. © 2018 University of Houston. All rights reserved.

Houston Journal of Mathematics

A class of sub-hardy hilbert spaces associated with weighted shifts

Let F and G be two bounded operators on two Hilbert spaces. Let their numerical radii be no greater than one. This note investigates when there is a Γ-contraction (S, P) such that F is the fundamental operator of (S, P) and G is the fundamental operator of (S*, P*). Theorem 1 puts a necessary condition on F and G for them to be the fundamental operators of (S, P) and (S*, P*) respectively. Theorem 2 shows that this necessary condition is also sufficient provided we restrict our attention to a certain special case. The general case is investigated in Theorem 3. Some of the results obtained for Γ-contractions are then applied to tetrablock contractions to figure out when two pairs (F1, F2) and (G1, G2) acting on two Hilbert spaces can be fundamental operators of a tetrablock contraction (A, B, P) and its adjoint (A*, B*, P*) respectively. This is the content of Theorem 3. © 2015 Elsevier Inc.

Admissible fundamental operators

We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz basic sequences.

Indiana University Mathematics Journal

The feichtinger conjecture and reproducing kernel hilbert spaces

We prove two theorems. The first theorem reduces to a scalar situation the well known vector-valued generalization of the Helson-Lowdenslager theorem that characterizes the invariant subspaces of the operator of multiplication by the coordinate function z on the vector-valued Lebesgue space L2(T ℂn). Our approach allows us to prove an equivalent version of the vector-valued Helson-Lowdenslager theorem in a completely scalar setting, thereby eliminating the use of range functions and partial isometries. The other three major advantages provided by our characterization are: (i) we provide precise necessary and sufficient conditions for the presence of reducing subspaces inside simply invariant subspaces; (ii) we give a complete description of the wandering vectors; (iii) we prove the theorem in the setting of all the Lebesgue spaces Lp (0 &lt; p ≤ ∞). Our second theorem generalizes the first theorem along the lines of de Branges' generalization of Beurling's theorem by characterizing those Hilbert spaces that are simply invariant under multiplication by zn and which are contractively contained in Lp (1 ≤ p ≤ ∞). This also generalizes a theorem of Paulsen and Singh [Proc. Amer. Math. Soc. 129 (2000)] as well as the main theorem of Redett [Bull. London Math. Soc. 37 (2005)]. © Instytut Matematyczny PAN, 2010.

Studia Mathematica

A finite multiplicity Helson-Lowdenslager-de Branges theorem

We give a short direct proof of Agler's factorization theorem that uses the Blecher-Ruan-Sinclair characterization of operator algebras. The key ingredient of this proof is an operator algebra factorization theorem. Our proof provides some additional information about these factorizations in the case of polynomials. © 2009 American Mathematical Society.