tailieunhanh - Báo cáo toán học: "Sub-Jordan operators: Bishop's theorem, spectral inclusion, and spectral sets "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Sub-Jordan nhà khai thác: Đức Giám Mục của định lý, bao gồm quang phổ, và quang phổ bộ. | J. OPERATOR THEORY 7 1982 373-395 Copyright by INCREST 1982 SUB-JORDAN OPERATORS BISHOP S THEOREM SPECTRAL INCLUSION AND SPECTRAL SETS JIM AGLER 0 INTRODUCTION In this paper we are interested in the following condition on an operator J e the algebra of bounded linear transformations on a complex Hilbert space Jf. There exist N Q 6 such jV ộ N is normal QN NQ and Qn 0. An operator J e is a Jordan operator if it satisfies for some positive integer n. We say e is an n-c Jordan operator if J satisfies with IIỔII c- The collection of n-c Jordan operators on a space will be denoted X . e is sub n-c Jordan if there is a Hilbert space and J e Xc such that is invariant for J and s J I . The collection of sub n-c Jordan operators on a space will be denoted by sub Xc . By a Jordan block we mean any operator which is unitarily equivalent to an operator of the form N c N 0 c N 0 c N n on JI where Ji is a Hilbert space of arbitrary dimension NeSỰỈ is normal and 0 c e R. If J is a Jordan block of the form 2 then the order of J 13-1789 374 JIM agler. is n and the constant of J is c. We set g jf equal to the set of Jordan blocks in with order n and constant c. Finally sub is the set of s e JSf p such that there exists a Hilbert space X and a J e ẩ X such that is invariant for J and s J 3 . We can now state the principal result of this paper. Recall that the strong topology of which we will denote in the sequel with the letter s is that topology defined by the family of seminorms px i XE where px T IITxl for Te jf . Theorem a. X -O s sub pr subỔ C f ố xr -s. Setting n 1 or c 0 reduces Theorem A to Bishop s Theorem for subnormal operators the strong closure of the normal operators on a Hilbert space is the subnormals on that space see 4 . The sub X sub part of Theorem A represents a generalization of the equivalence of conditions 2 and 3 in Theorem of 1 . As a byproduct of the proof of Theorem A we obtain the appropriate extension to the class of sub-Jordan .