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371.
Gen Qi Xu 《Mathematische Nachrichten》2023,296(6):2626-2656
In this paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, , that is, called the resolvent family for the linear evolution process with memory, the is called the memory effect family. In this paper, we prove that the families and are exponentially bounded, and the family associate with an operator pair that is called generator of the resolvent family. Using , we derive associated differential equation with memory and representation of via L. These results give necessary conditions of the well-posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on . The obtained results can be directly applied to linear delay differential equation, integro-differential equation and functional differential equations. 相似文献
372.
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
373.