Hille-yosida theorems for local convoluted C-semigroups and cosine functions

Abstract

The theory of convoluted C-operator families is an active research field. The main purpose of this paper is to prove several Hille-Yosida type theorems for local convoluted C-semigroups and cosine functions.

Spectral inclusions of exponentially bounded C -semigroups

“…Conversely, if A is a closed linear operator which commutes with C and if the ACP for A has a unique solution u(t, x) such that \| u(t, x)\| \leq M e at \| C - 1 x\| for every initial value x \in D(A), then A generates an exponentially bounded C-semigroup, under some additional conditions on A. One excellent source for studying C-semigroups is [7]. For other results concerning this theory we refer to [3,4,9,10,12,14].…”

Multiplicative perturbations of local C-cosine functions

“…Moreover, a local C-cosine function is not necessarily extendable to the half line [0, ∞) (see [22]) except for C = I (identity operator on X). Perturbations of local C-cosine functions with or without the exponential boundedness have been extensively studied by many authors appearing in [2,6,[8][9][10][11][12][13][14][15][16][17]19,23,25]. Some interesting applications of this topic are also illustrated there.…”

On mixed C-semigroups of operators on Banach spaces

“…A C-cosine operator family gives the solution of a well-posed Cauchy problem. For more details on the theory of cosine operator function we refer to [8,10,13,17].…”

“…With C = I, the identity operator on X, the C-semigroup {T(t)} t≥0 is said to be a C 0 -semigroup. Regularized semigroups and their connection with the ACP(A; x) have been studied, e.g., in [2,11,13,16].…”