TY - JOUR

T1 - Note on some operator equations and local spectral properties

AU - An, Il Ju

AU - Ko, Eungil

N1 - Funding Information:
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2009-0083521). This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2009-0093827).
Publisher Copyright:
© 2016, Element D.O.O. All Rights Reserved.

PY - 2016/6

Y1 - 2016/6

N2 - In this paper we define Sk, j by the set of solutions (A,B) of the operator equations AkBj+1Ak = A2k+j and BkAj+1Bk = B2k+j. Then we observe the set Sk, j is increasing for all integers k ≥ 1 and j ≥ 0. Now let a pair (A,B) ∈ Sk, j ∩ Sj+1,k−1 for any integer k ≥ 1 and j ≥ 0. We show that if any one of the operators A, AB, BA, and B has Bishop’s property (β), then all others have the same property. Furthermore, we prove that the operators Ak+j, AkBj+1, Aj+1Bk, Bj+1Ak, BkAj+1 and Bk+j have the same spectra and spectral properties. Finally, we investigate their Weyl type theorems.

AB - In this paper we define Sk, j by the set of solutions (A,B) of the operator equations AkBj+1Ak = A2k+j and BkAj+1Bk = B2k+j. Then we observe the set Sk, j is increasing for all integers k ≥ 1 and j ≥ 0. Now let a pair (A,B) ∈ Sk, j ∩ Sj+1,k−1 for any integer k ≥ 1 and j ≥ 0. We show that if any one of the operators A, AB, BA, and B has Bishop’s property (β), then all others have the same property. Furthermore, we prove that the operators Ak+j, AkBj+1, Aj+1Bk, Bj+1Ak, BkAj+1 and Bk+j have the same spectra and spectral properties. Finally, we investigate their Weyl type theorems.

KW - Operator equations

KW - Single valued extension property

KW - Spectrum

UR - http://www.scopus.com/inward/record.url?scp=84975246089&partnerID=8YFLogxK

U2 - 10.7153/oam-10-22

DO - 10.7153/oam-10-22

M3 - Article

AN - SCOPUS:84975246089

SN - 1846-3886

VL - 10

SP - 397

EP - 417

JO - Operators and Matrices

JF - Operators and Matrices

IS - 2

ER -