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Construction of strong derivable maps via functional calculus of unbounded spectral operators in Banach spaces

Benedetto Silvestri

Abstract

We provide sufficient conditions for the existence of a strong derivable map and calculate its derivative by employing a result in our previous work on strong derivability of maps arising by functional calculus of an unbounded scalar type spectral operator $R$ in a Banach space and the generalization to complete locally convex spaces of a classical result valid in the Banach space context. We apply this result to obtain a sequence of integrals converging to an integral of a complete locally convex space extension of a map arising by functional calculus of $R$.

Construction of strong derivable maps via functional calculus of unbounded spectral operators in Banach spaces

Abstract

We provide sufficient conditions for the existence of a strong derivable map and calculate its derivative by employing a result in our previous work on strong derivability of maps arising by functional calculus of an unbounded scalar type spectral operator in a Banach space and the generalization to complete locally convex spaces of a classical result valid in the Banach space context. We apply this result to obtain a sequence of integrals converging to an integral of a complete locally convex space extension of a map arising by functional calculus of .
Paper Structure (4 sections, 3 theorems, 29 equations)

This paper contains 4 sections, 3 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. notation
  3. Results
  4. Data availability statement and declaration

Key Result

Theorem 2

Let $K\subseteq\mathbb{R}$ be a compact nondegenerate interval of $\mathbb{R}$, and $U$ be an open neighbourhood of $\sigma(R)$ such that $K\cdot U\subseteq U$. Furthermore let $X$ be a compact space, $\mathfrak{g}\in\mathcal{C}(X\times K,\mathfrak{L}_{E}^{\infty}(U))$, $\{x_{n}\}_{n\in\mathbb{N}}$ and for every $n\in\mathbb{N}$ and $t\in K$ we have Then there exist $y\in X$ and a subsequence $\

Theorems & Definitions (8)

  • Definition 1
  • Theorem 2
  • proof
  • Definition 3
  • Lemma 4
  • proof
  • Corollary 5
  • proof