Embedding operators into strongly continuous semigroups

Abstract

Abstract. We study linear operators T on Banach spaces for which there exists a C0-semigroup (T (t)) t≥0 such that T = T (1). We present a necessary condition in terms of the spectral value 0 and give classes of examples where this can or cannot be achieved.

A "typical" contraction is unitary

“…For some classes of operators this question has a positive answer, e.g., for operators with spectrum in a certain area using functional calculus, see e.g. Haase [6, Section 3.1], and for isometries on Hilbert spaces with infinite-dimensional kernel, see [2].…”

On typical properties of Hilbert space operators

“…We obtain the following results. In Section 3, we recall some results obtained by the first author (see [8], [10]) about typical properties in the weak topology. In this topology, a typical contraction is unitary, it has maximal spectrum and empty point spectrum, it can be embedded into a C 0 -semigroup, and typical contractions are not unitarily equivalent.…”

Embeddability of real and positive operators

“…It is clear that L is a real contraction. We adapt the construction of the semigroup from [8,Prop. 4.3].…”

“…An analogous question in stochastics and measure theory was considered in [16, Chapter III] and [12]. The operator theoretic setting was discussed in [14,15,8]. Moreover, see [5,29] for an analogous question in quantum information theory as well as [33,37,20] for applications of Markov embedding to sociology, biology and finance, respectively.…”

“…Example 2.6 (Spectrum of a real-embeddable operator). Let K ⊂ C be a compact set with K = K. We construct a real-embeddable operator T with σ(T ) = K. The construction is a natural modification of [8,Example 2.4. ].…”