ON LOG-HYPONORMAL OPERATORS K()TARO TANAHASHILet T E B(H) be a bounded linear operator on a complex Hilbert space H.
T E B(H) is called a log-hyponormal operator if T is invertible and log (TT*) <_ log (T'T).Since log: (0, oo) -+ (-0% oo) is operator monotone, for 0 < p _< 1, every invertible p-hyponormal operator T, i.e., (TT*) p <_ (T'T) p, is log-hyponormal. Let T be a log-hyponormal operator with a polar decomposition T = UIT[. In this paper, we show that the aluthge transform 2P = Irlsgt:rl r is ~-hyponorma]. Moreover, if meas(cr(T)) = 0, then T is normal. Also, we make a log-hyponormal operator which is not p-hyponormal for any 0 < p.
Abstract. Let T be a bounded linear operator on a Hilbert space. Among other things, it is shownand |T * 2 | |T * | 2 , and (3) if T and T * are w -hyponormal, and either ker T ⊆ ker T * or ker T * ⊆ ker T , then T is normal.Mathematics subject classification (1991): 47B20, 47A63. Key words and phrases: Hyponormal operator, p -hyponormal operator, log-hyponormal operator, w -hyponormal operator, paranormal operator.
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