A Hilbert Space Problem Book

Spectral bounds for the independence ratio and the chromatic number of an operator

“…We recall some definitions and facts from Hilbert space theory. For background we refer the reader to the Hilbert space problem book [18] by Halmos. Let (V, Σ, µ) be a measure space consisting of a set V , a σ-algebra Σ on V , and a measure µ. We consider the Hilbert space…”

Operators with a given part of the numerical range

“…However, if H is not finite-dimensional, then the numerical range is not closed, in general. For instance, the numerical range of the backward shift on 2 is the open unit disk D(0, 1). If A * ∈ B(H ) is the adjoint operator of A ∈ B(H ), then W (A * ) = { λ; λ ∈ W (A)}.…”

Bounds on the 𝐿² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality

“…Finally, we note that the sesquilinear form associated with the operator J can be written as (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) (…”

“…For any bounded linear operator T on a complex Hilbert space, the numerical range W(T) is defined to be the set of all numbers (f,TF) as / ranges over unit vectors. Recall the following facts [8,9]:…”

“…This sesquilinear form can be written in a form more closely resembling (2.13)-(2.15) by introducing an enlarged state space S* = S U {00} (where 00 ^ S). We then define, for any function / on S, the extended function /* on S* by ", x f f(x) HxES, (3)(4)(5)(6)(7)(8) f…”