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Gaps between Operator Norm and Spectral and Numerical Radii of the Tensor Product of Operators

Zameddin I. Ismailov*, Pembe Ipek Al

Karadeniz Technical University, Department of Mathematics, 61080, Trabzon, Turkey

Abstract: One of the fundamental problem of the Spectral Theory of Linear Operators is to determine of the geometric place of the spectrum of the given operator and calculate the spectral and numerical radii of this operator. Other important problem in this theory is to explained the situation the spectral (numerical) radius is equal or not to operator norm. The only known way to calculate the spectral radius to date is the classical Gelfand formula, which often presents great technical challenges. Also, there is not yet a method of calculating the numerical radius for an operator. It should be noted that the finding the numerical range and numerical radius means maximally localizing the spectrum of an operator. The main purpose of this paper is to determine the relations gaps between operator norm and spectral and numerical radii of the tensor product operators associated with the compatible gaps of coordinate operators.

Keywords: operator norm, spectral radius, numerical radius

Received August 28, 2021; Revised October 02, 2021; Accepted October 11, 2021

Citation: Zameddin I. Ismailov, Pembe Ipek Al, Gaps between Operator Norm and Spectral and Numerical Radii of the Tensor Product of Operators. Turkish Journal of Analysis and Number Theory, 9 (2). 22-24. January 2021. doi:10.12691/tjant-9-2-1

1. Introduction

As is known from the mathematical literature, in a Banach space one of the most fundamental questions of the Spectral Theory of Linear Operators is to clearly determine the geometric location of the spectrum or resolvent sets of the operator and calculate the spectral radius of the operator. Although in many cases, serious theoretical and technical difficulties are encountered in finding the spectrum set of nonselfadjoint linear operators, as it is known from the literature, the only formula used in the spectral radius of linear bounded operators calculation is Gelfand formula. Let be a Banach space and be a linear bounded operator in , i.e. , Gelfand formula is as follows:

[1]. As it is known . Let be a linear bounded normal operator in a Hilbert space . It is true the relation (see [2]). But generally, these equations may not be provided for non-normal linear bounded operators. For example, the gap between operator norm and spectral radius of the operator is Moreover, let be a quasinilpotent operator. It is clear that The determination and definition the specific sub-classes of the operators family defined by is theoretically important as well as providing great benefits in applied mathematics.

Let be a Hilbert space and It is known that

[3]. If is a normal operator, then (see [4]). But this inequality may not be true for non-normal operators. Generally, for the following inequality is true (see [5,6]). One of the most fundamental questions is to identify specific subclasses of the family of operators that satisfy the condition.

In this study, the spectral and numerical gap characteristics of the tensor product of linear bounded operators defined on the tensor product of a finite number of Hilbert spaces are expressed with the appropriate characteristic numbers of the coordinate operators that form the tensor product.

Demuth's open problem in 2015 had a great impact on the emergence and shaping of the subject examined in this article (see [7]).

2. Gap between Operator Norm and Spectral Radius of the Tensor Product of Operators

In this section, gap between operator norm and spectral radius of the tensor product of operator (for more information of tensor product see [8]) will be investigated. It is known that for an operator the spectral radius is defined as

(see [5]). Namely, it is true the following result.

Theorem 1 Let be two separable Hilbert spaces and For the tensor product

is valid

Proof

By the definition

it is known that

From [9], for the spectrum of it is known that

Hence,

Corollary 1 Under the assumptions of the last theorem it is true

1. ,

2.

Example 1 Let and In this case, and Then, and Consequently, by Theorem 1 it is obtain that

Corollary 2 By using the last theorem for the tensor product

of the operators is true

Generally, using the Theorem 1 it can be proved the following proposition.

Corollary 3 For the tensor product

of the operators it is true

Corollary 4 From the Corollary 3 we have

(1)

(2) if and only if

(3) in special case when and , then

(4) in special case when

then in order to the necessary and sufficient condition is

3. Gap between Operator Norm and Numerical Radius of the Tensor Product of Operators

In this section, the gap between operator norm and numerical radius of the tensor product of operators associated with the gap of coordinate operators will be investigated.

Definition 1 [3] The numerical range and the numerical radius of the linear bounded operator in any Hilbert space define by

and

respectively.

It is easy to prove the following proposition.

Theorem 2 For the numerical range and numerical radius of tensor product operator of the operators in tensor product of Hilbert spaces are true.

and

where denotes the convex hull of the set .

Note that the analogous types result of Theorem 1 are true in this case too. For example, it is true the following result.

Theorem 3 For the tensor product of the operators in the tensor product of Hilbert spaces the numerical gap is of the form

Example 2 Let and Volterra integration operator in In this case, we get

and from [5] and [10]

Consequently, by Theorem 3 we have

References

[1]   | I. M. Gelfand, Normierte ringe, Matematicheskii Sbornik, 9, 51 (1941), 3-24.
[2]   | F. Hirsh, G. Lacombe, Elements of functional analysis, Springer, New York, 1999.
[3]   | K. E. Gustafson, D. K. M. Rao, Numerical Range: The field of values of linear operators and matrices, Springer, New York, 1997.
[4]   | J. H. Anderson, C. Foias, Properties which normal operators share with normal derivations and related operators, Pacific Journal of Mathematics, 61, 2, 1975, 313-325.
[5]   | P. R. Halmos, A Hilbert space problem book, Van Nostrand, New York, 1967.
[6]   | M. H. Stone, Linear transformations in Hilbert Space and their applications to analysis, Amer. Math. Soc. Colloquium Publications, 5, New York, 1932.
[7]   | M. Demuth, Mathematical aspect of physics with non-selfadjoint operators, List of open problem, American Institute of Mathematics Workshop, 8-12 June 2015.
[8]   | Y. M. Berezanskii, Z. G. Sheftel, G. F. Us, Functional analysis II, Birkhauser Verlag, Basel Switzerland, 1990.
[9]   | A. Brown, C. Pearchy, Spectra of tensor products of operator, Proceedings of the American Mathematical Society 17, 1, 1966, 162-166.
[10]   | M. Gürdal, M. T. Garayev, S. Saltan, Some concrete operators and their properties, Turkish Journal of Mathematics, 39, 2015, 970-989.