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Norm estimates for functions of non-selfadjoint operators nonregular on the convex hull of the spectrum

Gil’ M.

Demonstratio Mathematica

2017

Vol. 50, nr 1

267--277

We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A−A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An(n = 1, 2,...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.

On the normality of first-order differential operators

Ismailov Z.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 2

139-145

We investigate the relationship between the normality property of a first-order differential operator in a Hilbert space of vector-functions from the interval [O, 1] into a separable Hilbert space and the operator coefficients of the differential-operator expression which generates this operator.

Non-linearity of the set of p-quasi-analytic vectors for some essentially self-adjoint operators

Rusinek J.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 3

287-292

We show that, for any p [is greater than or equal to] 1, there exists an essentially self-adjoint operator for which the set of p-quasi-analytic vectors is not linear.

A nonstandard difference Sturm-Liouville operator

Javorskij J., Ljance V.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 3

285-290

After Nelson's Radically Elementary Probability Theory [1] a natural question arises: whether a hyperfinite-dimensional space is sufficiently rich to be used for the same goal as an infinite-dimensional one. Here a hyperfinite 3-diagonal matrix is investigated, which spectral properties are simular to the Naimarks's singular nonselfadjoint Sturm-Liouville differential operator on semi-axis [2, 3].