Strongly Integrable Operator-Valued Functions, Generated Vector Measures and Compactness of Integrals

TL;DR

This paper investigates the conditions under which integrals of operator-valued functions are compact and establishes new inequalities and approximation results, especially when the underlying space lacks an isomorphic copy of ℓ¹.

Contribution

It proves that integrals of compact operator families are compact under certain conditions and generalizes spectral radius inequalities for commuting families.

Findings

Integrals of compact operator families are compact if X does not contain ℓ¹.

Established a spectral radius inequality for commuting operator families.

Provided approximation results in spaces with finite dimensional Schauder decompositions.

Abstract

Gel'fand integral of a family of compact operators on a Hilbert space is not always compact, even with additional property of positivity and commutativity. We prove that integrals of a family, consisting of compact operators, in the space $L_{s}^1(\Omega,\mu,\mathcal{B}(X, Y))$ of strongly integrable families are compact whenever $X$ does not contain an isomorphic copy of $\ell^1$. In addition, we prove an integral inequality for spectral radius $$r\left(\int_\Omega\mathscr{A} \,d\mu\right)\leqslant\int_\Omega r(\mathscr{A}_t)\,d\mu(t)$$ for a mutually commuting family $\mathscr{A}$ in $L_s^1(\Omega,\mu,\mathcal{B}(X))$, which generalizes a recent result obtained under a stronger assumption of Bochner integrability. We prove also approximation results in $L_s^1(\Omega,\mu,\mathcal{B}(X))$ in the case $X$ has finite dimensional Schauder decomposition. All these results are based on a key theorem of this paper which states that every function in $L_{s}^1(\Omega,\mu, \mathcal{B}(X, Y))$ generates a countably additive, in operator norm, $\mathcal{B}(X, Y)$-valued measure whenever $X^*$ does not contain an isomorphic copy of $c_0$.