In this paper, we investigate the conditions under which some classes of operators in a complex Hilbert space H are said to be consistent in invertibility. AMS subject classification: 47B47, 47A30, 47B20 Keywords and phrases: Quasi-invertibility, Consistent in Invertibility, Quasinormal, Projection

Stephen, Luketero

Wabuya, Kikete

English

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Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.8, No.7, 2018  49  SOME RESULTS ON OPERATORS CONSISTENT IN INVERTIBILITY  Luketero Stephen and Kikete Wabuya*  School of Mathematics, University of Nairobi, P.O. Box 30197 – 00100 Nairobi, Kenya.  Abstract In this paper, we investigate the conditions under which some classes of operators in a complex Hilbert space H are said to be consistent in invertibility. AMS subject classification: 47B47, 47A30, 47B20 Keywords and phrases: Quasi-invertibility, Consistent in Invertibility, Quasinormal, Projection.  1. INTRODUCTION  In this paper, Hilbert spaces or subspaces will be denoted by capital letters, 𝐻 and 𝐾 respectively and , , ,T S A B etc. denotes bounded linear operators where an operator means a bounded linear transformation, (𝐻) will denote the Banach algebra of bounded linear operators on 𝐻. B(𝐻, 𝐾) denotes the set of bounded linear transformations from 𝐻 to 𝐾, which is equipped with the (induced uniform) norm. If 𝑇∈B(𝐻), then 𝑇* denotes the adjoint while Ker(T  ) denotes the kernel of T . For an operator 𝑇, we also denote by (𝑇) the spectrum of T.   An operator 𝑇∈(𝐻) is said to be: - Invertible if it has zero kernel - Quasi-invertible if it is injective and has a dense range - Positive if 0T    - Projection if 2T T   - Normal if * *T T TT   - Quasinormal if * *T TT TT T   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.8, No.7, 2018  50  - Consistent in invertibility (C.I) if both TS and ST  are either invertible or non-invertible together. 2. RESULTS Theorem 2.1 Let ( )T B H . If KerT  = 0 = Ker *T , then T  is a C.I operator. Proof If KerT  = 0, we have that T is invertible, it follows that *T is also invertible.  Since *TT  is a product of invertible operators it has to be invertible too. We also have that * *( )TT is invertible. But * *( )TT =*T T . Thus both *TT and *T T are invertible together. Hence T is a C.I operator.  Corollary 2.2 Let ( )T B H be quasi-invertible. Then T is a C.I operator. Proof If T is quasi-invertible, it follows that it is injective and has a dense range. As a consequence of being injective, we have that KerT  = 0 therefore T is a C.I operator.  Corollary 2.2 Let * ( )T B H  be such that *0 ( )W T . Then *T is a C.I. operator. Proof Recall that * *( ) ( )T W T   Therefore *0 ( )W T  *0 ( )T 0 in not an eigenvalue of *T  *T is invertible  *T is a C.I. operator.  Theorem 2.3 Let , ( )A B B H  be normal operators and * *AB B A , then A iB  is a C.I. operator. Proof * * * * * *( ) ( )AB B A AB B A   i.e. * *BA A B . It is enough to show that A iB is normal. * * *( )A iB A iB    Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.8, No.7, 2018  51  * * ** * * ** * * ** * * *( ) ( ) ( )( )                                                      ( ) ( )                           ( ) ( )...............(i)A iB A iB A iB A iBA A iA B iB A B BA A B B i A B B AA A B B i BA AB               * * ** * * ** * * *( )( ) ( )( )                                                      ( ) ( )..............(ii)A iB A iB A iB A iBAA iAB iBA BBAA BB i BA AB           From (i) and (ii) above it follows that A iB is normal, hence a C.I. operator.  Theorem 2.4 Let , , ( )A B X B H satisfy the operator equation AXB X where X is a quasi-invertible operator. Further, let A and B be quasinormal operators, then A and *B are C.I. operators. Proof Since A  is quasinormal, we have * * 0A AA AA A  . By the hypothesis that AXB X it follows that:   * ** ** ** *           since                  since  has a dense rangeAA AXB AA XA AAXB AA XA AX AA X AXB XA A AA X   Therefore, A is a normal operator, hence consistent in invertibility. It can similarly be shown that *B is consistent in invertibility.   REFERENCES 1. J. B. Conway, A course in Functional analysis,Springer-Verlag, New York, 1985 2. M. Hladnik and M. Ombialic, Spectrum of the product of operators, Proc. Amer. Math. Soc., 102 (1988), 300 – 302 3. P. Y. Wu, Product of normal operators, Canad. J. Math. 40 (1988), 1322 – 1330 4. W. Gong and D. Han, Spectrum of the product of operators and compact perturbations, Proc. Amer. Math. Soc., 120 (1994), 755 – 760. 

SOME RESULTS ON OPERATORS CONSISTENT IN INVERTIBILITY

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SOME RESULTS ON OPERATORS CONSISTENT IN INVERTIBILITY

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