Grothendieck operators on tensor products

Tensor Products and Grothendieck Type Inequalities of Operators in ¿„-spaces Bernd Carl and Andreas Defant

Several results in the theory of (p, £?)-summing operators are improved by a unified but elementary tensor product concept. Introduction Since the pioneering work of Grothendieck in Linear Functional Analysis there is an extensive literature dealing with operators in Lp-spaces. Still the most important result in this direction is Grothendieck's integral characterization [9] of operators from l\...

Spectra of Tensor Products of Operators

p-Summing Operators on Injective Tensor Products of Spaces

Let X, Y and Z be Banach spaces, and let ∏ p(Y, Z) (1 ≤ p < ∞) denote the space of p-summing operators from Y to Z. We show that, if X is a £∞-space, then a bounded linear operator T : X⊗̂ǫY −→ Z is 1-summing if and only if a naturally associated operator T : X −→ ∏ 1(Y, Z) is 1-summing. This result need not be true if X is not a £∞-space. For p > 1, several examples are given with X = C[0, 1] t...

Tensor Products of Closed Operators on Banach Spaces

Let A and B be closed operators on Banach spaces X and Y. Assume that A and B have nonempty resolvent sets and that the spectra of A and B are unbounded. Let 01 be a uniform cross norm on X @ Y. Using the Gelfand theory and resolvent algebra techniques, a spectral mapping theorem is proven for a certain class of rational functions of A and B. The class of admissable rational functions (includin...

Resolvents of operators on tensor products of Euclidean spaces

We consider the operator T = m k=1 A 1k ⊗ A 2k (1 ≤ m < ∞), where A lk are n l × n l matrices (k = 1,. .. , m; l = 1, 2), ⊗ means the tensor product. Norm estimates for the resolvent of that operator are derived. By these estimates, we obtain bounds for a solution X of the equation m k=1 A 1k X A 2k = C and explore perturbations of that equation. The norm estimates for the resolvent of T enable...