Operator Functions and Operator Equations

The following sections are included:

• Preface
• Contents

This chapter is of a preliminary character. Here we accumulate some well-known results on operators and operator-valued functions. The material of this chapter is systematically used in the remaining chapters of the book.

This chapter is devoted to representations of solutions to some classes of operator equations in Banach spaces.

Throughout this chapter 𝒳 and 𝒴 are Banach spaces with the unit operators I_𝒳 = I and I_𝒴 = I, respectively. Recall that ℬ(𝒴;𝒳) means the set of all bounded operators acting from 𝒴 into 𝒳 and ℬ(𝒳) = ℬ(𝒳, 𝒳)…

This chapter is devoted to functions of finite dimensional operators. In particular, we establish norm estimates for the resolvents and functions, regular on the convex hull co(A) of the spectrum. We also estimate the functions of the form $\frac{1}{f}(A),$ where f is regular on co(A). By these estimates in the next chapter we investigate matrix equations. In addition, spectrum perturbations of matrices are explored.

In this chapter we consider the Sylvester, quasi-Sylvester, polynomial and generalized polynomial equations in Euclidean spaces. Solution estimates in terms of the spectral (operator) norm are established. We also derive norm estimates for a class of functions of two matrix arguments.

Throughout this chapter $A\in C^{n\times n},B\in C^{\tilde{n}\times \tilde{n}}$ and $C\in C^{n\times \tilde{n}}(n,\tilde{n}<\infty )$.

In this chapter we consider the two-sided Sylvester equation

An eigenvalue is said to be simple, if its geometric multiplicity is equal to one. In this chapter we consider a matrix A whose all the eigenvalues are simple. As it is well known, in this case there is an invertible matrix T, such that

This chapter is devoted to functions of some classes of compact operators in a Hilbert space. In particular, we establish norm estimates for the resolvent of a Schatten-von Neumann operator and for functions of a Hilbert-Schmidt operator, regular on the convex hull of the spectrum. Applications to spectrum perturbations of compact operators and to operator equations whose coefficients are compact operators are also discussed.

The present chapter deals with non-compact non-normal operators A in a separable Hilbert space ℋ. Besides, it is supposed that either A*–A ∈ SN_p or A*A–I ∈ SN_p (p ≥ 1). We suggest the triangular representations of the considered operators via chains of their invariant projections. These representations are our main tool in the next two chapters devoted to operator functions of non-selfadjoint operators and the relevant operator equations.

All the operators considered in this chapter are bounded operators in ℋ.

In the present chapter we derive norm estimates for the resolvents of bounded non-selfadjoint operators A. It is supposed that either A has a Schatten-von Neumann Hermitian component, or A*A – I is a nuclear operator. We also suggest bounds for the non-unitary eigenvalues of A, provided A*A – I is compact.

In this chapter we establish the norm estimates for functions of a bounded operator A in a Hilbert space ℋ, assuming that the functions are regular on the convex hull of the spectrum of A. Besides, it is assumed that either A–A* is a Hilbert-Schmidt operator, or AA*–I is a nuclear one. Applications of the obtained estimates to the Sylvester equation are also discussed. In addition, we investigate the rotations of invariant subspaces of operators with compact Hermitian components.

This chapter deals with some functions of an unbounded operator in a separable Hilbert space ℋ. In particular, we establish sharp norm estimates for the resolvent and function e^At (t ≥ 0) of an unbounded operator A with a compact Hermitian component. In addition, a norm estimate for the resolvent of an unbounded operator inverse to a Schatten-von Neumann one is derived. By that estimate we investigate the Hirsch functions of A. Fraction powers and operator logarithm are the examples of the Hirsch functions.

As in the finite dimensional case an operator A in a separable Hilbert space ℋ is said to be a diagonalizable operator, if there are a boundedly invertible operator T ∈ ℬ(ℋ) and a normal operator D acting in ℋ, such that

In this chapter we consider some unbounded diagonalizable operators and derive bounds for the condition numbers of the considered operators. We also discuss applications of the obtained bounds to spectrum perturbations and operator functions.

Let 𝒳 be a Banach space. For A, B, Ã ∈ ℬ(𝒳), [A, B] := AB – BA is the commutator, [A, B, Ã] := AB – BÃ is the generalized commutator; [f(A), B] := f(A)B–Bf(A) and [f(A), B, f(Ã)] := f(A)B–Bf(Ã) will be called the function commutator and the generalized function commutator, respectively.

In this chapter we derive norm estimates for $[f\left(A\right),B,f\left(\tilde{A}\right)]$. In the case B = I these estimates give us the bounds for $\parallel f\left(A\right)-f\left(\tilde{A}\right)\parallel$.

This chapter is devoted to a class of functions of two non-commuting operator arguments. For these functions, norm estimates are derived. Applications of the obtained estimates to the generalized polynomial equations in Hilbert spaces are also discussed.

The following sections are included:

• Bibliography
• List of Symbols
• Index