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| Analytic Operator Rings and Analytic J-Self Adjoint Operator Ring |
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Citation： |
Ma Jipu.Analytic Operator Rings and Analytic J-Self Adjoint Operator Ring[J].Chinese Annals of Mathematics B,1982,3(5):595~598 |
| Page view： 944
Net amount： 701 |
Authors： |
Ma Jipu; |
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| Abstract： |
Let A be a linear bounded operator in Hilbert space H with polar respresentation A =J(A^*A)^{1/2} where J^2=I, J^* = J. we use \pho_J(A) to denote the set of all complex \lambda, such that for any $\lambda \in \rho_J(A)$ there exist an bounded inverce R_J(A,\lambda) of (A—\lambda J)and \sigma_J(A)to complement of \rho_J(A).
Let S be a closed Cauchy domain, S\supset \sigma_J(A) and f (z) an analytic function on S. We define
$f(A)=\frac{1}{2\pi i}\oint\limits_{2s} {f(\zeta ){R_J}(A,\zeta )d\zeta }$,
the set of all such f(A)is denoted M.
If f(z) be analytic on S and symmetrical for real axis then f(A) is J-self adjoint. The set of all such f(A)is denoted M'. Let A\otimes B = AJB for A, B \in M(or M'). We have
Theorem, the ring of functions analytic on S (or analytic symmetrical for real axis on S) is a algebra homomorphism of M (or M'). The constant function 1 or z corresponds to operator J or A^* respectively.
Let $M_J={JB|B \in M}$ and $M'_J={JB|B \in M'}$ If the spectrum of (A^*A)^{1/2} is detached, we have
Theorem. M_J has common non-trivial reducing subspace and it is true for M_J. |
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