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Characterizations of subnormal operators. (English) Zbl 0731.47020

Let \({\mathcal S}=(S_ 1,...,S_ d)\) be a d-tuple of commuting operators on a Hilbert space H. \({\mathcal S}\) is said to be subnormal if there exists a d-tuple \({\mathcal N}=(N_ 1,...,N_ d)\) of commuting normal operators on some Hilbert space \(K\supset H\) such that \(S_ j\subset N_ j\), \(j=1,...,d\). For \(a=(a_ 1,...,a_ d)\in {\mathfrak N}^ d\), write \({\mathcal S}^ a=S_ 1^{a_ 1}S_ 2^{a_ 2}...S_ d^{a_ d}\). The author improves the Halmos-Bram-Itô criterion for subnormality as follows:
If X is a linear subspace of H such that \(H=\bigvee \{{\mathcal S}^ aX:\) \(a\in {\mathfrak N}^ d\}\) and that \(\sum_{a,b}\sum_{p,q}({\mathcal S}^{b+p}f,S^{a+q}f)z(p,q)\overline{z(a,b)}\geq 0\) for every \(f\in X\) and every finitely non-zero function z(.,.) from \({\mathfrak N}^ d\times {\mathfrak N}^ d\) to \({\mathfrak C}\), then \({\mathcal S}\) is subnormal. For the case where \(S_ j\), \(j=1,...,d\), are contractions, the author proves that \({\mathcal S}\) is subnormal if \(\sum_{a,b}({\mathcal S}^{a+b}f,{\mathcal S}^{a+b}f)z(b)\overline{z(a)}\geq 0\) for every \(f\in H\) and every finitely non-zero function z(.) from \({\mathfrak N}^ d\) to \({\mathfrak C}\) with \(\sum_{a}z(a)=0.\)
Some characterizations of algebraic normal operators are obtained.

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
15B48 Positive matrices and their generalizations; cones of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
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