$$({P},\;m)$$ ( P , m ) - B -normal and quasi- P
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Advances in Operator Theory https://doi.org/10.1007/s43036-020-00118-x
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ORIGINAL PAPER
ðP, mÞ-B-normal and quasi-P-B-normal operators in semi-Hilbert spaces Naeem Ahmad1 Received: 18 June 2020 / Accepted: 17 October 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, we introduce a new family of operators which is called polynomiallym-B-normal (resp-quasi polynomially B-normal). Some of the basic properties of members of these families are studied. Keywords Semi-inner product A-normal operator Polynomially normal (n, m)-normal
Mathematics Subject Classification 47B99 47A05
1 Introduction and preliminaries Let K; k:kÞ be a complex infinite dimension Hilbert space and let Lb ðKÞ denote the C -algebra of all bounded linear operators on K and let Lb ðKÞþ be the set of positive operators of Lb ðKÞ defined as B
if positive if and only if
hBa j ai 0 8
a 2 K:
For R 2 Lb ðKÞ, we denote range R by rangðRÞ, the nullspace by kerðRÞ and by R the adjoint operator of R. If R 2 Lb ðKÞ, then rangðRÞ stands for its closure in the norm topology of K. We denote by QrangðBÞ the orthogonal projection onto the closed linear subspace rangðBÞ Communicated by S. Djordjevic. & Naeem Ahmad [email protected]; [email protected] 1
Mathematical Analysis and Applications Mathematics Department, College of Science, Jouf University, P.O.Box 2014, Sakaka, Saudi Arabia
N. Ahmad
If B 2 Lb ðKÞþ , the map W : K K ! C define a positive semi-definite sesquilinear form given by Wða; bÞ ¼ ha j biB where ha j biB ¼ hBa j bi: Note that Wða; aÞ ¼ 0 () hBa j ai ¼ 0 () kak2B ¼ 0 () a 2 kerðBÞ: It easily to check that the map W defines a semi-inner product on K, and the seminorm induced by the semi-inner product is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 kakB ¼ ha j aiB ¼ k Bak2 ; for each a 2 K: Clearly, kakB ¼ 0 if and only if a 2 kerðBÞ. An operator R 2 Lb ðKÞ is called B-bounded if there exists a [ 0 such that kRakB akakB ; 8 a 2 K: The set of all B-bounded operators on K will be denoted by LBb ðKÞ. For any R 2 LBb ðKÞ, we set from [2] kRakB kRkB :¼ sup ; a 2 rangðBÞ : kakB Definition 1.1 ([2]) Let R; S 2 Lb ðKÞ. S is called an B-adjoint operator of R if R and S satisfy hRa j biB ¼ ha j SbiB ; 8 a; b 2 K; or equivalently BR ¼ S B. In particular, R is called an B-selfadjoint operator If R is an B-adjoint of itself or equivalently BR ¼ R B: By Douglas Theorem ([10, 13]), an operator R 2 LðKÞ admits an B-adjoint if and only if rangðR BÞ rangðBÞ: In the following, the set of all operator R 2 Lb ðKÞ which admit an B-adjoint is denoted by LbB ðKÞ: Throughout this paper, B always denotes a positive operator in Lb ðKÞ with rangðBÞ ¼ rangðBÞ. In [2] it was observed that if R 2 LbB ðKÞ, the solution of the equation BX ¼ R B is a distinguished B-adjoint operator of R, which is denoted by R] or RB . However, RB is given by RB ¼ By R B where By is the Moore-Penrose inverse of B. The B-adjoint operator RB verifies 8 BRB ¼ R B > > > > > > < rangðRB Þ rangðBÞ >
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