摘要
<正> 设H是复Hilbert空间,B(H)为H上有界线性算子全体所成的Banach代数。对于T∈B(H),如果T~*T的迹tr(T~*T)<∞,则称T为Hilbert-Schmidt算子,其全体记为,它是B(H)中的双边理想。
An operator T acting on a Hilbert space H is called k- quasihyponormal if {T} =T*k (T*T-TT*)Tk≥0 . Let A and B be two operators on H, one can obtain an operator τ =τAB on the class (?)2 of all Hilbert-Schmidt operators on H in such a way that τ(X)=AXB for every X∈(?)2. In this note the authors show thatLemma Assume that T is a k-quasihyponormal operator on H and Tk≠ 0 . Let {T} =(?)0∞λdEλ be the spectral decomposition of {T}. Given ε> 0 , denoteHε=(?)0εdEλH. For any non-negative integers m and n, write T*mTn =with . Then we must have ‖Smn‖≥rm+n, where r is the spectral radius of T.Theorem τ is a k- quasihyponormal operator on the Hilbert-Schmidt class (?)2 if and only if one of the following conditions holds: ( 1 ) Both A and B* are k-quasihyponormal ;( 2 ) Ak = 0 or Bk = 0 .Theorem Suppose that Ak≠0 and Bk≠ 0 . Then both A and B* are k- qua-sihyponormal operators if and only if the inequality({A}XBk+1x, XBk+1x)≥(- {B*}X*A*kAx, X*A*kAx) holds for every rank- one operator X and every vector x in H.
