| Xia Daoxing,Li Shaokuan.[J].数学年刊A辑,1980,1(3-4):501~504 |
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| ON FUNCTIONAL TRANSFORMATION OF THESEMI-HYPONORMAL OPERATORS |
| Received:December 10, 1979 |
| DOI： |
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| Let H be a complex separable Hilbert space. If T=UP is a bounded operator and
\(P - PU{P^ * } = D(T) \ge 0\), where we always Suppose that U is nnitary and \(P \ge 0\)，then T is called semi-hypormal. For the bounded closed set E of the real line R we set
\（S(E) = \left\{ {\left. \varphi \right|{K_\varphi }({x_1},{x_2}) = \frac{{\varphi ({x_1}) - \varphi ({x_2})}}{{{x_1} - {x_2}}}{\rm{is semipositive kernel of an integral operator in }}{L^2}(E)} \right\}\）, For the closed set E of the unit circle G±, we set \[\psi '(E) = \left\{ {\left. \varphi \right|{H_\varphi }(\xi ,\eta ) = \frac{{1 - \varphi (\xi )\bar \varphi (\eta )}}{{1 - \xi \bar \eta }} is semi-positive kernel of an integral operator in {L^2}(E)} \right\}\],We haye proved
Theorem 1. Let T = UP be semi-hyponormal, \(\varphi \in \psi '(\sigma (U))\)，then \(\tilde T = \varphi (U)\) is also semi-Hyponormal；
Theorem 2. Let T=UP be semi-hyponormal operator, \(\psi \in S([0,\left\| T \right\|])\) and \(\psi \) be
positive valued. Then \(\tilde T = U\psi (P)\) is also semi-hyponormal.
Theorem 3. Let T = UP be Semi-hypormal operator and \(\psi \) be soalar-funotion on
\([0,\infty )\) .If \(\tilde T = U\psi (P)\) is а]яо Semi-hypomormal, then
we have \[\begin{array}{l}
\sigma (\tilde T) = {l_\psi }(\sigma (T)),\\left\| {D(T)} \right\| \le \frac{1}{\pi }{\psi ^{ - 1}}(r)d\theta
\end{array}\]
Theorem 4. Let T=UP be semi-hypormal operator and soalar-funotion \(\psi \in S([0,\left\| T \right\|])\)，then\(\tilde T = U\psi (P)\)is semi-hypormal and (1), (2)are valid. |
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