Abstract

In the present study, firstly, some algebraic inequalities are proved, which will be used later. By making use of these relations, some evaluations are found related the gaps between norm and numerical radius, spectral radius and Crawford number for diagonal block operator matrices on the infinite direct sum of Hilbert spaces. Later on, the gaps between some spectral characteristic numbers (operator norm, lower and upper bounds of spectrum set and numerical radius) of the infinite direct sum of Hilbert space operators relatively to the same spectral characteristics of the coordinate operators are investigated. Then, the obtained results are supported by applications. The open problem posed by Demuth in 2015 and the works of Kittaneh and his researcher group in this area had an important effect on the formation of the subject discussed in this study.

1. Introduction

As is known in the mathematical literature, obtaining the spectrum set, numerical range set of a given operator and and calculating spectral radii, numerical radii and Crawford number is one of the fundamental questions of the spectral theory of linear operators. Generally, finding the spectrum set and numerical range of non-selfadjoint linear bounded operators is theoretically and technically quite difficult. For the calculation of the spectral radius of the linear bounded operator in any Banach space there is one formula (see ¹⁰). On the other hand, for the spectral and the numerical radius, the following inequalities hold and for .

In addition, for the linear normal bounded operator in Hilbert space we have the following relations

It is beneficial to recall that for the spectrum set and numerical range of any linear bounded operator the following spectral inclusion holds (see ^{8, 10} for more information).

In ¹³, some spectral radius inequalities for block operator matrix, sum, product, and commutators of two linear bounded Hilbert space operators have been examined.

In ¹, some estimates for numerical and spectral radii of the Frobenius companion matrix have been obtained.

In ², some upper and lower bounds for the numerical indices in Hilbert space operators have been found.

In ³, some estimates have been obtained for spectral and numerical radii of the product, sum, commutator, anticommutator of two Hilbert spaces operators.

In ⁷, several numerical radius inequalities have been proved for block operator matrices in the direct sum of Hilbert spaces. In ⁵, the numerical radius inequalities have been given in for accretive matrices.

In ⁴, several new norms and numerical radius inequalities have been researched for block operator matrices.

In ¹⁴, several new -numerical radius inequalities have been offered for many type block operator matrices in the direct sum of Hilbert spaces.

In ¹⁹, subadditivity of the spectral radius of commutative two operators in Banach spaces has been investigated. In ²¹, by the same author the subadditivity and submultiplicativity properties of local spectral radius of bounded positive operators have been researched in Banach spaces. In ²⁰, the same properties of local spectral radius in partially ordered Banach spaces have been established. In ²², several inequalities for the spectral radius of a positive commutator of positive operators have been surveyed in Banach space ordered by a normal and generating core. In ^{9, 12}, the numerical range and numerical radius of some Volterra integral operator in Hilbert Lebesgue spaces at finite interval have been considered.

The open problem posed by Demuth in 2015 and the works of Kittaneh and his researcher group in this area had an important effect on the formation of the subject discussed in this study (see, e.g. ^{6, 13, 14}).

This paper is organized as follows: The first section is devoted to introduction proving. We contrive by the necessary auxiliary theorem in Section 2. In the last section, we prove our main results. Also, the obtained results are supported by applications.

2. Some Auxiliary Important Results

In this section, we will prove certain auxiliary results from which it will be used later.

From ¹¹ and ¹⁷, we have the following theorem.

Theorem 1 Let . For the numbers and , the inequalities

are true.

Proof. For all cases, we prove by mathematical induction method.

For , it is clear that

Now, assume that

for any , .

Then, one can easily have

From this and by mathematical induction method, for any the following inequality hold

Similarly, for by simple calculations we again have that

Now, assume that for ,

From this assumption, one can have

Consequently, by mathematical induction method it is obtained

for any .

Now, we prove the second part of the theorem.

For it is clear that

Then, we have

On the other hand, we get

Then, for , we have

Now, assume that the mentioned inequalities hold for . Then for , it is clear that

On the other hand, we have

Theorem 2 For the sequences of real numbers and the following inequalities hold

and

Proof. For the prove this claims, it will be used of Theorem 1.

For any , we have

Then, from the last equation we have

For any , it is true

From the last relation, it is obtained

On the other hand, from the following simple calculations we have

Then, from this inequality, we have

Later on, from the Theorem 1 it is obtained that for any

Hence,

3. Some Relations between Spectral Characteristic Numbers of Infinite Direct Sum of Hilbert Space Operators and Their Coordinate Operators

Let and be the spectrum and numerical range sets of the linear bounded operator A in any Hilbert space H, respectively (see ⁸). Also, assume that

and

It is well known that for any (for more information see ^{8, 10}).

In addition, let is a Hilbert space, for and

Remember that some connections between some spectral characteristic numbers of the direct sum of Hilbert space operators with same numbers of coordinate operators have been investigated in ^{16, 18}.

Using Theorem 2, the following results can be proved.

Theorem 3 For the direct sum of operators in the following inequalities hold

and

where, and for

Proof. From the ¹⁵, we know

Also, from the ¹⁶, we know

and

Consequently,

Now assumed that, firstly

secondly

and thirdly

In this case, apply the Theorem 2 in according places it will be obtained validity of the claims of theorem, respectively.

Now, it will be proved the following result.

Theorem 4 Let If are smallest numbers satisfy the following conditions

and

respectively, then

Proof. It is clear that

Then, from ¹⁵ and ¹⁶

From this and since m is the smallest, it is obtained that

On the contrary, since is a smallest number satisfying the condition

using the technique in proof of Theorem 2 the following inequalities hold

Then, at least one of

Consequently, it is implies that

Hence,

Example Let

In this case, we have

and

If we have and Hence, If we have and Hence,

Let

and

We have,

and

Hence,

In the direct sum of Banach spaces , consider the following operator in the form

where, and .

It can be verified that

and

Then, we have

where,

References