arXiv subjects

Michael Cwikel

Publications and source records attributed to Michael Cwikel.

24 records · Page 2Linked to original sources

arXiv2009

Homotonic Algebras

An algebra $\mathcal{A}$ of real or complex valued functions defined on a set $\mathbf{T}$ shall be called \textit{homotonic} if $\mathcal{A}$ is closed under forming of absolute values, and for all $f$ and $g$ in $\mathcal{A}$, the product $f\times g$ satisfies $|f\times g|\le|f|\times|g|$. Our main purpose in this paper is two-fold: To show that the above definition is equivalent to an earlier definition of homotonicity, and to provide a simple inequality which characterizes sub-multiplicativity and strong stability for weighted sup norms on homotonic algebras.

math.RA↗

arXiv2008

Estimates for covering numbers in Schauder's theorem about adjoints of compact operators

Let T:X --> Y be a bounded linear map between Banach spaces X and Y. Let S:Y' --> X' be its adjoint. Let B(X) and B(Y') be the closed unit balls of X and Y' respectively. We obtain apparently new estimates for the covering numbers of the set S(B(Y')). These are expressed in terms of the covering numbers of T(B(X)), or, more generally, in terms of the covering numbers of a "significant" subset of T(B(X)). The latter more general estimates are best possible. These estimates follow from our new quantitative version of an abstract compactness result which generalizes classical theorems of Arzela-Ascoli and of Schauder. Analogous estimates also hold for the covering numbers of T(B(X)), in terms of the covering numbers of S(B(Y')) or in terms of a suitable "significant" subset of S(B(Y')).

math.FA↗

arXiv2008

Complex interpolation of compact operators mapping into lattice couples

Suppose that (A_0,A_1) and (B_0,B_1) are Banach couples, and that T is a linear operator which maps A_0 compactly into B_0 and A_1 boundedly (or even compactly) into B_1. Does this imply that T maps [A_0,A_1]_s to [B_0,B_1]_s compactly for 0<s<1 ? (Here, as usual, [A_0,A_1]_s denotes the complex interpolation space of Alberto Calderon.) This question has been open for 44 years. Affirmative answers are known for it in many special cases. We answer it affirmatively in the case where (A_0,A_1) is arbitrary and (B_0,B_1) is a couple of Banach lattices having absolutely continuous norms or the Fatou property. Our result has some overlap with a recent result by Evgeniy Pustylnik.

math.FA↗

arXiv2006

Complex interpolation of compact operators mapping into the couple (FL^{\infty},FL_{1}^{\infty})

If (A_0,A_1) and (B_0,B_1) are Banach couples and a linear operator T from A_0 + A_1 to B_0 + B_1 maps A_0 compactly into B_0 and maps A_1 boundedly into B_1, does T necessarily also map [A_0,A_1]_s compactly into [B_0,B_1]_s for s in (0,1)? After 42 years this question is still not answered, not even in the case where T is also compact from A_1 to B_1. But affirmative answers are known for many special choices of (A_0,A_1) and (B_0,B_1). Furthermore it is known that it would suffice to resolve this question in the special case where (B_0,B_1) is the special couple (l^\infty(FL^\infty), l^\infty(FL^\infty_1)). Here FL^\infty is the space of all sequences which are Fourier coefficients of bounded functions, FL^\infty_1 is the weighted space of all sequences (a_n) such that (e^n a_n) is in FL^\infty, and thus B_0 and B_1 are the spaces of bounded sequences of elements in these spaces (i.e., they are spaces of doubly indexed sequences). We provide an affirmative answer to this question in the related but simpler case where (B_0,B_1) is the special couple (FL^\infty,FL^\infty_1).

math.FA↗

arXiv2005

K-divisibility constants for some special couples

We prove new estimates of the $K$-divisibility constants for some special Banach couples. In particular, we prove that the $K$-divisibility constant for a couple of the form $(U\oplus V, U)$ where $U$ and $V$ are non-trivial Hilbert spaces equals $2/\sqrt{3}$. We also prove estimates for the $K$-divisibility constant of the two-dimensional version of the couple $(L_2,L_\infty)$, proving in particular that this couple is not exactly $K$-divisible. There are also several auxiliary results, including some estimates for relative Calderón constants for finite dimensional couples.

math.FA↗

arXiv1992

Interpolation of compact operators by the methods of Calderón and Gustavsson-Peetre

Let $ X=(X_0,X_1)$ and $ Y=(Y_0,Y_1)$ be Banach couples and suppose $T: X\to Y$ is a linear operator such that $T:X_0\to Y_0$ is compact. We consider the question whether the operator $T:[X_0,X_1]_θ\to [Y_0,Y_1]_θ$ is compact and show a positive answer under a variety of conditions. For example it suffices that $X_0$ be a UMD-space or that $X_0$ is reflexive and there is a Banach space so that $X_0=[W,X_1]_α$ for some $0<α<1.$

math.FA↗