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Near subnormal operators and subnormal operators. (English) Zbl 0702.47012
A Hilbert space operator $$A\in B(H)$$ is called a D-near subnormal operator if $$D\geq M^*ADA$$ for some positive $$D\in B(H)$$ and some positive constant M. In case $$D=A^*A-AA^*\geq 0$$ and A is D-near subnormal, A is called a near subnormal operator. It is shown that every subnormal operator is near subnormal but the converse is false. Also, every hyponormal operator is trivially near subnormal but, again, the converse is false. The example of a nonsubnormal operator which is near subnormal is based on a criterion of subnormal operators which is the main result of the paper. The criterion involves sequences $$D_ n$$ and $$B_ n$$ of operators such that $D_ 0=A^*A-AA^*,\quad B_ 0=A,\quad B_ 1=D_ 0^{1/2} B_ 0D_ 0^{+1/2},\quad B_ n=D^{1/2}_{n-1} B_{n-1}D^{+1/2}_{n-1},\quad D_ n=Q_{B_ n}+D_{n-1},$ where $$D^{+1/2}$$ denotes the Moore-Penrose inverse of $$D^{1/2}$$.