Spectral theory for contraction semigroups on Hilbert space

Exponential Decay of Semigroups in Hilbert Space

We give simple bounds for the exponential decay of a strongly continuous semigroup V (t) = exp(At) in a Hilbert space. The bounds are expressed by means of the solution X of the Ljapunov equation A X + XA = ?I and the quantity sup 0th kV (t)k for small h. Our main bound is attained on any normal semigroup and can therefore not be improved in general. Let V (t) be a strongly continuous semigroup...

Spectral Theory of Operator Measures in Hilbert Space

In §2 the spaces L2(Σ,H) are described; this is a solution of a problem posed by M. G. Krĕın. In §3 unitary dilations are used to illustrate the techniques of operator measures. In particular, a simple proof of the Năımark dilation theorem is presented, together with an explicit construction of a resolution of the identity. In §4, the multiplicity function NΣ is introduced for an arbitrary (non...

Elementary Hilbert Space Theory

A Hilbert Space is an inner product space that is complete. Let H be a Hilbert space and for f, g ∈ H, let 〈f, g〉 be the inner product and ‖f‖ = √ 〈f, f〉. (1.1) We will consider Hilbert spaces over C. Most arguments go through for Hilbert spaces over R, and the arguments are simpler. We assume that the reader is familiar with some of the basic facts about the inner product. Let us prove a few t...

OPERATOR THEORY ON HILBERT SPACE Class notes

Game Theory Formulated on Hilbert Space

We present a consistent formulation of quantum game theory that accommodates all possible strategies in Hilbert space. The physical content of the quantum strategy is revealed as a family of classical games representing altruistic game play supplemented by quantum interferences. Crucial role of the entanglement in quantum strategy is illustrated by an example of quantum game representing the Be...