tailieunhanh - Báo cáo toán học: "An irreductible subnormal operator with infinite multiplicities "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một nhà điều hành dưới nhiệt độ irreductible với multiplicities vô hạn. | Copyright by INCREST 1985 J. OPERATOR THEORY 13 1985 291- 297 AN IRREDUCIBLE SUBNORMAL OPERATOR WITH INFINITE MULTIPLICITIES J. B. CONWAY and c. R. PUTNAM 1. HYPONORMAL OPERATORS Let H be a separable infinite dimensional Hilbert space and let the operator T on H with the Cartesian representation TA Í5 be completely hyponormal. Thus T T - TT f D 0 and T has no normal part. Let A Re T have the spectral resolution A ị tdEt. If p is any Borel set on the real line and if F t denotes the linear measure of the vertical cross section ơ 7j n z Re z t of ơ T then 1-2 n E P DE P n Also if n t denotes the spectral multiplicity function of A then n tr E p DE p a ị í r F r dr. See 7 and 8 . See also 4 p. 539 for an inequality related to . In particular relation implies that tt Z Ị meas20 T . 2. INFINITE MULTIPLICITIES The following is a variation of . Theorem 1. Tf T A B is completely hyponormal then 7ĩmaxơe F t dr P X 292 J. B. CONWAY and c. R. PUTNAM where ơfD is the essential spectrum of D and Poo denotes the set of points where A Re T has infinite multiplicity. Proof If a denotes the set of points where A has finite multiplicity it follows from and that a Z 1 2 hence also E a DE 0 is compact see 3 also 9 Choose a sequence a- of unit vectors converging weakly to 0 and satisfying D1 2 max ffe D 1 2 x - 0. The arrow will denote strong convergence unless otherwise indicated. Since a E Poo I then max ơe Z lim II D1I2X II2 lim supịlE à Dll2x II2 lim supllE Poo Dll2xnII2 lim sup 0O 1 2x 2 E p D by that is . Clearly Theorem 1 can be regarded as a refinement of the result 3 that D is compact whenever A has only finite multiplicities. Theorem 2. Let T A ÌB be completely hyponormal and suppose that D eafD and that equality holds in thus ơ Z meas2ơ . Then E Poo Ỉ so that A has uniformly infinite multiplicity. A similar assertion holds also for aA bB whenever a b are real and a2 b2 0. Proof. As above let a denote the points where A