Inequalities of Hardy-Littlewood-Polya type for functions of operators and their applications
In this paper, we derive a generalized multiplicative Hardy-Littlewood-Polya type inequality, as well as several related additive inequalities, for functions of operators in Hilbert spaces. In addition, we find the modulus of continuity of a function of an operator on a class of elements defined with the help of another function of the operator. We then apply the results to solve the following problems: (i) the problem of approximating a function of an unbounded self-adjoint operator by bounded operators, (ii) the problem of best approximation of a certain class of elements from a Hilbert space by another class, and (iii) the problem of optimal recovery of an operator on a class of elements given with an error.
Keywords:Inequalities of Hardy-Littlewood-Polya type functions of operatorsmodulus of continuity best approximation of unbounded operatorsoptimal recovery of operators
Msc:MSC 26D10 MSC 47A63 MSC 41A17 MSC 47A58
1 Definitions, notation, and statements of main problems
Let be Banach spaces and be an operator (not necessarily linear) with domain . Let . We define the modulus of continuity of the operator on the class to be
The problem of computing the modulus of continuity of an operator on the given class of elements is an abstract version of the problem of finding sharp Landau-Kolmogorov type inequality (see, for instance, (babenko_inequalities, Ch. 7)).
Let be a set of linear bounded operators from to with norms bounded by The quantity
is called the deviation of the operator from the operator on the class . Finally,
is called the best approximation of operator by a set of bounded operators on the class
Stechkin’s problem (see, for instance, arestov_approx, stechkin_inequalities, stechkin_best_approx, and babenko_inequalities, Ch. 7.1) of the best approximation of the operator on class consists of computing and finding (studying the questions of its existence, uniqueness, characterization) the “extremal” operator, i.e. the one that delivers in (1).
Let be the set of all mappings of the space into the space , be the set of all linear operators from into , and be the set of all bounded linear operators into .
For and operator we set
The problem of optimal recovery of the operator with the help of the set of mappings (recovery methods) on the elements from the class , given with error , consists of finding the quantity and operator , which realizes .
Let be a set in and The quantity
is called the best approximation of the element by the set
Let be convex classes in and is a real number. The set is called a homothet of the class with the homothety coefficient The quantity
is called the best approximation of the class by a homothet
The problem of best approximation of the class by a homothet consist of computing the quantity (2).
All above listed problems are closely related to Landau-Kolmogorov type inequalities (see, for instance, arestov_approx, babenko_inequalities §§7.3 - 7.5). We need the following two theorems, which formally establish the connection.
(Stechkin stechkin_best_approx). Let be a homogeneous (in particular, linear) operator and let be a centrally symmetric subset of . Then
If is a centrally symmetric set and is a homogeneous operator, then
In this paper, we consider the above stated problems in the case when , where is a Hilbert space, the considered operators are some functions of self-adjoint operator in , and the class of elements is also defined with the help of some function of the same operator.
Let us mention some known results of this type for operators in Hilbert spaces.
First of all, let us mention the classical Hardy-Littlewood-Polya inequality hardy_inequalities for function from , such that the derivative of order in Sobolev sense also belongs to the space :
This inequality implies the following estimate for the modulus of continuity of the operator on the class :
In addition, the sharpness of Hardy-Littlewood-Polya inequality implies that in fact
Hardy-Littlewood-Polya inequalty and the above result on computing of the modulus of continuity have been generalized in multiple directions (see, for instance, babenko_inequalities, babenko_ligun_shumeiko, babenko_approx, Bab_Kryachko). In particular, in Bab_Kryachko sharp Hardy-Littlewood-Polya type inequality was proved for functions of operators with a discrete spectrum.
The problem of best approximation of the unbounded operator by bounded operators on the class was solved in subbotin_best_approx. It was proved there that
The result was further generalized babenko_approx to the case of higher degrees of self-adjoint operators in Hilbert spaces.
In subbotin_best_approx it was also proved that
This was generalized in bapprclasses to the case when classes are defined with the help of degrees of arbitrary self-adjoint operators.
The paper is organized as follows. In Section 2 we introduce the necessary definitions and facts from spectral theory of self-adjoint operators in Hilbert spaces. In particular, here we define functions of such operators. In Section 3 we obtain rather general Hardy-Littlewood-Polya type inequality for functions of unbounded self-adjoint operators. In addition, we find the modulus of continuity of a function of an operator on a class of elements defined with the help of another function of an operator. In Section LABEL:S4 we solve the problem of best approximation of a function of an unbounded self-adjoint operator by bounded operators. In Section LABEL:S5 we obtain a series of sharp additive Hardy-Littlewood-Polya type inequalities for functions of operators. The problem of approximation of one class of elements from a Hilbert space by another class is solved in Section LABEL:S6. Finally, in Section LABEL:S7 we solve the problem of optimal recovery of operators on a class of elements given with an error.
2 Preliminaries from Spectral Theory
We begin by reminding some necessary facts on operator Stieltjes integrals and functions of self-adjoint operators in Hilbert spaces.
Let be a Hilbert space with an inner product and norm We consider a linear unbounded operator in with domain .
First, let us recall some definitions and facts from spectral theory of self-adjoint operators (see, for instance, §75 and §88 in akhiezer_theory).
Partition of unity is a one parametric family of projection operators , defined on a finite or infinite interval (in the case when the interval is infinite, we understand, by definition,
in strong convergence sense) and satisfying the following properties:
b) in the sense of strong convergence
c) ( is an identity operator)
We set for and for
It follows from the definition that for any the quantity
is left-continuous, non-decreasing function of bounded variation for which
Thus, we have -measure that allows the construction of Lebesgue-Stieltjes integral.
If any condition is satisfied with respect to all -measures, generated by elements , then we say that it is satisfied with respect to the operator measure .
Now for the defined, measurable, and finite almost everywhere with respect to the operator measure functions, we may consider operator integrals (for details and properties of such integrals see, for instance, akhiezer_theory)
Based on the spectral theorem, each self-adjoint operator has a corresponding partition of unity , such that
In addition, element belongs to the domain of the operator if and only if
Moreover, if then
Let now function be defined, measurable, and finite almost everywhere with respect to the operator measure . We also assume that there exists a dense set in of elements , such that
Under the made assumptions, the function of an operator is an operator defined as follows
3 Inequalities of Hardy-Littlewood-Polya type and the problem of computing the modulus of continuity
We begin with the case when rather general Hardy-Littlewood-Polya type inequality can be proved in a simple and explicit manner. We consider functions and of an operator , where and are continuous complex-valued functions on such that and are even and strictly increasing on . In addition, we assume
where is a strictly increasing, concave function, and
A rather general Hardy-Littlewood-Polya type inequality is contained in the following theorem.
Let be an unbounded self-adjoint operator in . Then for any the following inequality holds
If, in addition, is such that
then the inequality (6) is exact.
In order to obtain the upper estimate, we apply Jensen’s inequality ( is a concave function and ). For any we obtain
The fact that under assumption (7) obtained inequality is sharp will follow from the next theorem.
By we denote the class of elements such that
Let be an unbounded self-adjoint operator in and let be the modulus of continuity of the operator on the class . Then for any
If, in addition, is such that the assumption (7) is satisfied, then for any