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A commutant lifting theorem on analytic polyhedra

Calin Ambrozie, Jörg Eschmeier (2005)

Banach Center Publications

In this note a commutant lifting theorem for vector-valued functional Hilbert spaces over generalized analytic polyhedra in ℂⁿ is proved. Let T be the compression of the multiplication tuple M z to a *-invariant closed subspace of the underlying functional Hilbert space. Our main result characterizes those operators in the commutant of T which possess a lifting to a multiplier with Schur class symbol. As an application we obtain interpolation results of Nevanlinna-Pick and Carathéodory-Fejér type...

A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

Let T be a linear operator on a Banach space X with s u p ₙ | | T ⁿ / n w | | < ∞ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) n - 1 ∑ k = 0 n - 1 T k converges uniformly; (ii) c l ( I - T ) X = z ∈ X : l i m n ∑ k = 1 n T k z / k e x i s t s .

A convex treatment of numerical radius inequalities

Zahra Heydarbeygi, Mohammad Sababheh, Hamid Moradi (2022)

Czechoslovak Mathematical Journal

We prove an inner product inequality for Hilbert space operators. This inequality will be utilized to present a general numerical radius inequality using convex functions. Applications of the new results include obtaining new forms that generalize and extend some well known results in the literature, with an application to the newly defined generalized numerical radius. We emphasize that the approach followed in this article is different from the approaches used in the literature to obtain such...

A counterexample to the Γ-interpolation conjecture

Adama S. Kamara (2015)

Annales Polonici Mathematici

Agler, Lykova and Young introduced a sequence C ν , where ν ≥ 0, of necessary conditions for the solvability of the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. They conjectured that condition C n - 2 is necessary and sufficient for the solvability of an n-point interpolation problem. The aim of this article is to give a counterexample to that conjecture.

A density theorem for algebra representations on the space (s)

W. Żelazko (1998)

Studia Mathematica

We show that an arbitrary irreducible representation T of a real or complex algebra on the F-space (s), or, more generally, on an arbitrary infinite (topological) product of the field of scalars, is totally irreducible, provided its commutant is trivial. This provides an affirmative solution to a problem of Fell and Doran for representations on these spaces.

A discrepancy principle for Tikhonov regularization with approximately specified data

M. Thamban Nair, Eberhard Schock (1998)

Annales Polonici Mathematici

Many discrepancy principles are known for choosing the parameter α in the regularized operator equation ( T * T + α I ) x α δ = T * y δ , | y - y δ | ≤ δ , in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and T * y δ are approximated by Aₙ and z ₙ δ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results are applicable...

A finite multiplicity Helson-Lowdenslager-de Branges theorem

Sneh Lata, Meghna Mittal, Dinesh Singh (2010)

Studia Mathematica

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(;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....

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