A factorization theorem for compact operators

Abstract

It is shown that every compact operator T:E-*F between Banach spaces admits a compact factorization (T=QP where P:E-+c and Q:c-*Fare compact) through a closed subspace c of the Banach space c0 of zero-convergent sequences.

Operator ideals and spaces of bilinear operators

“…With this theorem we are able to prove the following characterization of compact bilinear operators and this extends a characterization of compact linear operators due to Terzioglu [9]; see also Randtke [6]. THEOREM 2.7 For T E BL(X x Y; Z ) the following are equivalent 1" T is compact 2" There exist continuous bilinear functionals 4, on X x Y with lirnl/4,l/ = 0, such that for all (x, y) E X x Y…”

Quotient algebras of Banach operator ideals related to non-classical approximation properties

“…[8,Remark 11 (ii) Consider the factorisation T = BSA in part (i) of the quasi p-nuclear operator T in (3.9). According to the compact factorisation result of Terzioğlu [38] (or Randtke [31]), the compact operators A ∈ K(E, Y ) and B ∈ K(X, F ) admit compact factorisations A = A 2 A 1 and B = B 2 B 1 through closed subspaces M ⊂ c 0 and N ⊂ c 0 , respectively. Let U := B 1 SA 2 so that the following diagram commutes:…”

Closed ideals in the algebra of compact-by-approximable operators

“…In fact, if the closed subspace X ⊂ c 0 fails to have the A.P., then by [36,Theorem 1.e.4] there is a Banach space Y and an operator T ∈ K(Y, X) \ A(Y, X). By a compact factorisation theorem of Terzioğlu [52] (see also Randtke [45]) we may factor T compactly through a closed subspace of c 0 , that is, there is a closed subspace Z 0 ⊂ c 0 and A ∈ K(Y, Z 0 ), B ∈ K(Z 0 , X) such that T = BA. Consider Z = Z 0 ⊕ X ⊂ c 0 and define U ∈ L(Z) by U (x, y) = (0, Bx), (x, y) ∈ Z.…”