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A classification of projectors

Gustavo Corach, Alejandra Maestripieri, Demetrio Stojanoff (2005)

Banach Center Publications

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 .

Forms, functional calculus, cosine functions and perturbation

Wolfgang Arendt, Charles J. K. Batty (2007)

Banach Center Publications

In this article we describe properties of unbounded operators related to evolutionary problems. It is a survey article which also contains several new results. For instance we give a characterization of cosine functions in terms of mild well-posedness of the Cauchy problem of order 2, and we show that the property of having a bounded H -calculus is stable under rank-1 perturbations whereas the property of being associated with a closed form and the property of generating a cosine function are not....

Mazur-Orlicz equality

Fon-Che Liu (2008)

Studia Mathematica

A remarkable theorem of Mazur and Orlicz which generalizes the Hahn-Banach theorem is here put in a convenient form through an equality which will be referred to as the Mazur-Orlicz equality. Applications of the Mazur-Orlicz equality to lower barycenters for means, separation principles, Lax-Milgram lemma in reflexive Banach spaces, and monotone variational inequalities are provided.

On S. Mazur's problems 8 and 88 from the Scottish Book

V. V. Peller (2007)

Studia Mathematica

The paper discusses Problems 8 and 88 posed by Stanisław Mazur in the Scottish Book. It turns out that negative solutions to both problems are immediate consequences of the results of Peller [J. Operator Theory 7 (1982)]. We discuss here some quantitative aspects of Problems 8 and 88 and give answers to open problems discussed in a recent paper of Pełczyński and Sukochev in connection with Problem 88.

Quadratic functionals on modules over complex Banach *-algebras with an approximate identity

Dijana Ilišević (2005)

Studia Mathematica

The problem of representability of quadratic functionals by sesquilinear forms is studied in this article in the setting of a module over an algebra that belongs to a certain class of complex Banach *-algebras with an approximate identity. That class includes C*-algebras as well as H*-algebras and their trace classes. Each quadratic functional acting on such a module can be represented by a unique sesquilinear form. That form generally takes values in a larger algebra than the given quadratic functional...

Weighted projections into closed subspaces

G. Corach, G. Fongi, A. Maestripieri (2013)

Studia Mathematica

We study A-projections, i.e. operators on a Hilbert space 𝓗 which act as projections when a seminorm is considered in 𝓗. The A-projections were introduced by Mitra and Rao (1974) for finite-dimensional spaces. We relate this concept to the theory of compatibility between positive operators and closed subspaces of 𝓗. We also study the relationship between weighted least squares problems and compatibility.

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