Turkish Journal of Analysis and Number Theory
Volume 10, 2022 - Issue 1
Website: http://www.sciepub.com/journal/tjant

ISSN(Print): 2333-1100
ISSN(Online): 2333-1232

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Research Article

Open Access Peer-reviewed

Zameddin I. Ismailov^{ }, Pembe Ipek Al

Received November 14, 2022; Revised December 23, 2022; Accepted December 30, 2022

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.

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.

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,

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,

[1] | Abu-Omar, A., Kittaneh, F.: Estimates for the numerical radius and the spectral radius of the Frobenius companion matrix and bounds for the zeros of polyno- mials. Ann. Funct. Anal. 5(1), 56-62 (2014). | ||

In article | View Article | ||

[2] | Abu-Omar, A., Kittaneh, F.: Upper and lower bounds for the numerical radius with an application to involution operators. Rocky Mountain Journal of Math. 45(1), 1055-1064 (2015). | ||

In article | View Article | ||

[3] | Abu-Omar, A., Kittaneh, F.: Notes on some spectral radius and numerical in- equalities. Studia Math. 227(2), 97-109 (2015). | ||

In article | View Article | ||

[4] | Bani-Domi, W., Kittaneh, F.: Norm and numerical radius inequalities for Hilbert space operators. Linear and Multilinear Algebra 69(5), 934-945 (2021). | ||

In article | View Article | ||

[5] | Bedrani, Y., Kittaneh, F., Sabbabheh, M.: Numerical radii of accretive matrices. Linear and Multilinear Algebra 69(5), 957-970 (2021). | ||

In article | View Article | ||

[6] | Demuth, M.: Mathematical aspect of physics with non-selfadjoint operators, List of open problem. American Institute of Mathematics Workshop, Germany, 8-12 June 2015. | ||

In article | |||

[7] | Guelfen, H., Kittaneh, F.: On numerical radius inequalities for operator matrices. Numerical Functional Analysis and Optimization 40(11), 1231-1241 (2019). | ||

In article | View Article | ||

[8] | Gustafson, K. E., Rao, D. K. M.: Numerical Range: The Field Of Values Of Linear Operators And Matrices. Springer, New York (1997) | ||

In article | View Article | ||

[9] | Gu¨rdal, M., Garayev, M. T., Saltan, S.: Some concrete operators and their properties. Turkish Journal of Math. 39, 970-989 (2015). | ||

In article | View Article | ||

[10] | Halmos, P. R.: A Hilbert Space Problem Book. Van Nostrand, New York (1967). | ||

In article | |||

[11] | Ismailov, Z. I., O¨ ztu¨rk Mert, R.: Gaps between some spectral characteristics of direct sum of Hilbert space operators. Operators and Matrices 16(2), 337-347 (2022). | ||

In article | View Article | ||

[12] | Karaev, M. T., Iskenderov, N. Sh.: Numerical range and numerical radius for some operators. Linear Algebra and its Applications 432, 3149-3158 (2010). | ||

In article | View Article | ||

[13] | Kittaneh, F.: Spectral radius inequalities for Hilbert space operators. Proc. Amer. Math. Soc. 134(11) 385-390 (2006). | ||

In article | View Article | ||

[14] | Kittaneh, F., Sahoo, S.: On A-numerical radius equalities and inequalities for certain operator matrices. Ann. Funct. Anal. 12(52), 1-23 (2021). | ||

In article | View Article PubMed | ||

[15] | Naimark, M. A.: Continuous direct sums of Hilbert spaces and some of their ap- plications. Uspekhi Matematicheskikh Nauk 10, 111-142 (1955) (article in Russian). | ||

In article | |||

[16] | Otkun C¸ evik, E.: Some numerical characteristics of direct sum of operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69(2), 1221-1227 (2020). | ||

In article | |||

[17] | Otkun C¸ evik, E., Ismailov, Z. I.: Some gap relations between operator norm with spectral and numerical radii of direct sum Hilbert space operators. Lobachevskii Journal of Mathematics 43(2), 366-375 (2022). | ||

In article | View Article | ||

[18] | Otkun C¸ evik, E., Ismailov, Z. I.: Spectrum of the direct sum of operators. Elec- tronic Journal of Differential Equations 210, 1-8 (2012). | ||

In article | |||

[19] | Zima, M.: A theorem on the spectral radius of the sum of two operators and its application. Bull. Austral. Math. Soc. 48, 427-434 (1993). | ||

In article | View Article | ||

[20] | Zima, M.: On the local spectral radius in partially ordered Banach spaces. Czechoslovak Mathematical Journal 49(124), 835-841 (1999). | ||

In article | View Article | ||

[21] | Zima, M.: On the local spectral radius of positive operators. Proceedings of the American Mathematical Society 131(3), 845-850 (2003) | ||

In article | View Article | ||

[22] | Zima, M.: Spectral radius inequalities for positive commutators. Czechoslovak Mathematical Journal 64(139) 1-10 (2014). | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2022 Zameddin I. Ismailov and Pembe Ipek Al

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Zameddin I. Ismailov, Pembe Ipek Al. Some Spectral Characteristic Numbers of Direct Sum of Operators. *Turkish Journal of Analysis and Number Theory*. Vol. 10, No. 1, 2022, pp 21-26. http://pubs.sciepub.com/tjant/10/1/5

Ismailov, Zameddin I., and Pembe Ipek Al. "Some Spectral Characteristic Numbers of Direct Sum of Operators." *Turkish Journal of Analysis and Number Theory* 10.1 (2022): 21-26.

Ismailov, Z. I. , & Al, P. I. (2022). Some Spectral Characteristic Numbers of Direct Sum of Operators. *Turkish Journal of Analysis and Number Theory*, *10*(1), 21-26.

Ismailov, Zameddin I., and Pembe Ipek Al. "Some Spectral Characteristic Numbers of Direct Sum of Operators." *Turkish Journal of Analysis and Number Theory* 10, no. 1 (2022): 21-26.

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[1] | Abu-Omar, A., Kittaneh, F.: Estimates for the numerical radius and the spectral radius of the Frobenius companion matrix and bounds for the zeros of polyno- mials. Ann. Funct. Anal. 5(1), 56-62 (2014). | ||

In article | View Article | ||

[2] | Abu-Omar, A., Kittaneh, F.: Upper and lower bounds for the numerical radius with an application to involution operators. Rocky Mountain Journal of Math. 45(1), 1055-1064 (2015). | ||

In article | View Article | ||

[3] | Abu-Omar, A., Kittaneh, F.: Notes on some spectral radius and numerical in- equalities. Studia Math. 227(2), 97-109 (2015). | ||

In article | View Article | ||

[4] | Bani-Domi, W., Kittaneh, F.: Norm and numerical radius inequalities for Hilbert space operators. Linear and Multilinear Algebra 69(5), 934-945 (2021). | ||

In article | View Article | ||

[5] | Bedrani, Y., Kittaneh, F., Sabbabheh, M.: Numerical radii of accretive matrices. Linear and Multilinear Algebra 69(5), 957-970 (2021). | ||

In article | View Article | ||

[6] | Demuth, M.: Mathematical aspect of physics with non-selfadjoint operators, List of open problem. American Institute of Mathematics Workshop, Germany, 8-12 June 2015. | ||

In article | |||

[7] | Guelfen, H., Kittaneh, F.: On numerical radius inequalities for operator matrices. Numerical Functional Analysis and Optimization 40(11), 1231-1241 (2019). | ||

In article | View Article | ||

[8] | Gustafson, K. E., Rao, D. K. M.: Numerical Range: The Field Of Values Of Linear Operators And Matrices. Springer, New York (1997) | ||

In article | View Article | ||

[9] | Gu¨rdal, M., Garayev, M. T., Saltan, S.: Some concrete operators and their properties. Turkish Journal of Math. 39, 970-989 (2015). | ||

In article | View Article | ||

[10] | Halmos, P. R.: A Hilbert Space Problem Book. Van Nostrand, New York (1967). | ||

In article | |||

[11] | Ismailov, Z. I., O¨ ztu¨rk Mert, R.: Gaps between some spectral characteristics of direct sum of Hilbert space operators. Operators and Matrices 16(2), 337-347 (2022). | ||

In article | View Article | ||

[12] | Karaev, M. T., Iskenderov, N. Sh.: Numerical range and numerical radius for some operators. Linear Algebra and its Applications 432, 3149-3158 (2010). | ||

In article | View Article | ||

[13] | Kittaneh, F.: Spectral radius inequalities for Hilbert space operators. Proc. Amer. Math. Soc. 134(11) 385-390 (2006). | ||

In article | View Article | ||

[14] | Kittaneh, F., Sahoo, S.: On A-numerical radius equalities and inequalities for certain operator matrices. Ann. Funct. Anal. 12(52), 1-23 (2021). | ||

In article | View Article PubMed | ||

[15] | Naimark, M. A.: Continuous direct sums of Hilbert spaces and some of their ap- plications. Uspekhi Matematicheskikh Nauk 10, 111-142 (1955) (article in Russian). | ||

In article | |||

[16] | Otkun C¸ evik, E.: Some numerical characteristics of direct sum of operators. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69(2), 1221-1227 (2020). | ||

In article | |||

[17] | Otkun C¸ evik, E., Ismailov, Z. I.: Some gap relations between operator norm with spectral and numerical radii of direct sum Hilbert space operators. Lobachevskii Journal of Mathematics 43(2), 366-375 (2022). | ||

In article | View Article | ||

[18] | Otkun C¸ evik, E., Ismailov, Z. I.: Spectrum of the direct sum of operators. Elec- tronic Journal of Differential Equations 210, 1-8 (2012). | ||

In article | |||

[19] | Zima, M.: A theorem on the spectral radius of the sum of two operators and its application. Bull. Austral. Math. Soc. 48, 427-434 (1993). | ||

In article | View Article | ||

[20] | Zima, M.: On the local spectral radius in partially ordered Banach spaces. Czechoslovak Mathematical Journal 49(124), 835-841 (1999). | ||

In article | View Article | ||

[21] | Zima, M.: On the local spectral radius of positive operators. Proceedings of the American Mathematical Society 131(3), 845-850 (2003) | ||

In article | View Article | ||

[22] | Zima, M.: Spectral radius inequalities for positive commutators. Czechoslovak Mathematical Journal 64(139) 1-10 (2014). | ||

In article | View Article | ||