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A characterization of formally symmetric unbounded operators. (English) Zbl 0701.47010

The main result is: If A, B are symmetric operators on a Hilbert space, such that D(A)\(\subset D(B)\), \(D(A^*)\subset D(B^*)\) and \(\| (A^*-iB^*)x\| \leq \| A^*x\|\) on \(D(A^*)\), then \(B\subset 0\). Hence, criteria for an unbounded operator T to be formally symmetric (i.e. \(Tx=T^*x\) for x in \(D(T)\cap D(T^*))\), symmetric, or selfadjoint are obtained.
Reviewer: N.Angelescu

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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