Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Giá trị tuyệt đối của các nhà khai thác hoàn toàn hyponormal. | Ỉ. OPERATOR THEORY 8 1982 319 -326 Copyright by INCREST 1982 ABSOLUTE VALUES OF COMPLETELY HYPONORMAL OPERATORS c. R. PUTNAM 1. INTRODUCTION Only bounded operators on a separable Hilbert space H will be considered. If A is selfadjoint with the spectral resolution A ị tdE then A is said to be absolutely continuous if II EtxII2 is an absolutely continuous function of t for all X in H. A similar notion holds for unitary operators. An operator T A 4- ĨB A B self-adjoint is said to be hyponormal if T T TT DỳQ equivalently AB - BA. - ỈC ic D oi V 2 and completely hyponormal if in addition T has no normal part. See I for a recent survey of hyponormal operators. If T is completely hyponormal then A - - Re T as well as B Im T is absolutely continuous see 2 p. 42. Further in case T is completely hyponormal and has a polar factorization T - UP uunitary p IT -- rs T - then u is also absolutely continuous see 2 p. 21 and Lemma 4 of 4 In addition every absolutely continuous selfadjoint operator is the real part of some completely hyponormal operator and every absolutely continuous unitary operator is the unitary factor in the factorization of some completely hyponormal operator see 5 p. 323. Spectral properties of the absolute value Tị T T - of a completely hyponormal operator T are much less clear-cut. In the consideration of IT below it will always be supposed that T has a factorization . It turns out that a completely hyponormal T has such a unique factorization if and only if see the discussion in 3 . In case ơ U is a proper subset of the unit circle IT is surely 320 c. R. PUTNAM absolutely continuous 2 p. 21 but simple examples show that the latter assertion is not true in general. Thus for example if n t7 is the entire unit circle then ff Ti may consist of two points necessarily see below of infinite multiplicity in ơp TÍ . Even if ff CZ is the entire circle nevertheless under certain circumstances involving the spectrum of T T is absolutely continuous or .
