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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...

A unified analysis of elliptic problems with various boundary conditions and their approximation

Jérôme Droniou, Robert Eymard, Thierry Gallouët, Raphaèle Herbin (2020)

Czechoslovak Mathematical Journal

We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii)...

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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