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861.
862.
We show that if a densely defined closable operator A is such that the resolvent set of A2 is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial . We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that is normal for some , then T is normal. Hence a closed subnormal operator T such that is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that and are self-adjoint for some coprime numbers p and q, then A must be self-adjoint. 相似文献
863.
The present work is devoted to the construction of an asymptotic expansion for the eigenvalues of a Toeplitz matrix as goes to infinity, with a continuous and real-valued symbol having a power singularity of degree with , at one point. The resulting matrix is dense and its entries decrease slowly to zero when moving away from the main diagonal, we apply the so called simple-loop (SL) method for constructing and justifying a uniform asymptotic expansion for all the eigenvalues. Note however, that the considered symbol does not fully satisfy the conditions imposed in previous works, but only in a small neighborhood of the singularity point. In the present work: (i) We construct and justify the asymptotic formulas of the SL method for the eigenvalues with , where the eigenvalues are arranged in nondecreasing order and is a sufficiently small fixed number. (ii) We show, with the help of numerical calculations, that the obtained formulas give good approximations in the case . (iii) We numerically show that the main term of the asymptotics for eigenvalues with , formally obtained from the formulas of the SL method, coincides with the main term of the asymptotics constructed and justified in the classical works of Widom and Parter. 相似文献