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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 .

A note on generalized projections in c₀

Beata Deręgowska, Barbara Lewandowska (2014)

Annales Polonici Mathematici

Let V ⊂ Z be two subspaces of a Banach space X. We define the set of generalized projections by V ( X , Z ) : = P ( X , Z ) : P | V = i d . Now let X = c₀ or l m , Z:= kerf for some f ∈ X* and V : = Z l (n < m). The main goal of this paper is to discuss existence, uniqueness and strong uniqueness of a minimal generalized projection in this case. Also formulas for the relative generalized projection constant and the strong uniqueness constant will be given (cf. J. Blatter and E. W. Cheney [Ann. Mat. Pura Appl. 101 (1974), 215-227] and G. Lewicki...

Accretive approximation in C*-algebras

Reiner Berntzen (1996)

Studia Mathematica

The problem of approximation by accretive elements in a unital C*-algebra suggested by P. R. Halmos is substantially solved. The key idea is the observation that accretive approximation can be regarded as a combination of positive and self-adjoint approximation. The approximation results are proved both in the C*-norm and in another, topologically equivalent norm.

Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems

Jacek Jachymski (2009)

Studia Mathematica

Let X be a Banach space and T ∈ L(X), the space of all bounded linear operators on X. We give a list of necessary and sufficient conditions for the uniform stability of T, that is, for the convergence of the sequence ( T ) n of iterates of T in the uniform topology of L(X). In particular, T is uniformly stable iff for some p ∈ ℕ, the restriction of the pth iterate of T to the range of I-T is a Banach contraction. Our proof is elementary: It uses simple facts from linear algebra, and the Banach Contraction...

Convergence of orthogonal series of projections in Banach spaces

Ryszard Jajte, Adam Paszkiewicz (1997)

Annales Polonici Mathematici

For a sequence ( A j ) of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums S n = j = 1 n A j in a “strong” sense. Those limits turn out to be some special idempotent operators (unbounded, in general). In the case of X = L₂(Ω,μ), an arbitrary unbounded closed and densely defined operator A in X may be the μ-almost sure limit of S n (i.e. S n f A f μ-a.e. for all f ∈ (A)).

Explicit representation of compact linear operators in Banach spaces via polar sets

David E. Edmunds, Jan Lang (2013)

Studia Mathematica

We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. This provides a Banach space version of the well-known Hilbert space result of E. Schmidt.

Minimal multi-convex projections

Grzegorz Lewicki, Michael Prophet (2007)

Studia Mathematica

We say that a function from X = C L [ 0 , 1 ] is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Πₙ ⊂ X, where Πₙ denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general “shape” to preserve....

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