On an abstract evolution equation with a spectral operator of scalar type

Marat V Markin

doi:10.1155/s0161171202112233

Abstract

It is shown that the weak solutions of the evolution equation y′(t) = Ay(t), t ∈ [0, T) (0 < T ≤ ∞), where A is a spectral operator of scalar type in a complex Banach space X, defined by Ball (1977), are given by the formula y(t) = e tAf, t ∈ [0, T), with the exponentials understood in the sense of the operational calculus for such operators and the set of the initial values, f′s, being ∩ 0≤t<TD(e tA), that is, the largest possible such a set in X.

Highlights

Consider the evolution equation y (t) = Ay(t), t ∈ [0, T ) (0 < T ≤ ∞), (1.1)in a complex Banach space X with a spectral operator A of scalar type [2, 5]
It is readily seen that the notion of a weak solution of (1.1) is more general than that of the classical one
The purpose of the present paper is to stretch out [8, Theorem 3.1] which states that the general weak solution of (1.1) with a normal operator A in a complex Hilbert space is of the form y(t) = etAf, t ∈ [0, T ), f ∈

Summary

Introduction

In a complex Banach space X with a spectral operator A of scalar type [2, 5]. Following [1], by a weak solution of (1.1) with a densely defined linear operator A The purpose of the present paper is to stretch out [8, Theorem 3.1] which states that the general weak solution of (1.1) with a normal operator A in a complex Hilbert space is of the form y(t) = etAf , t ∈ [0, T ), f ∈

Objectives

Results

Conclusion