On unbounded integral operators with quasisymmetric kernels. (Russian. English summary) Zbl 1463.45058

Summary: In 1935 von Neumann established that a limit spectrum of self-adjoint Carleman integral operator in \(L_2\) contains 0. This result was generalized by the author on nonself-adjoint operators: the limit spectrum of the adjoint of Carleman integral operator contains 0. Say that a densely defined in \(L_2\) linear operator \(A\) satisfies the generalized von Neumann condition if 0 belongs to the limit spectrum of adjoint operator \(A^{\ast}\). Denote by \(B_0\) the class of all linear operators in \(L_2\) satisfying a generalized von Neumann condition. The author proved that each bounded integral operator, defined on \(L_2\), belongs to \(B_0\). Thus, the question arises: is an analogous assertion true for all unbounded densely defined in \(L_2\) integral operators? In this note, we give a negative answer on this question and we establish a sufficient condition guaranteeing that a densely defined in \(L_2\) unbounded integral operator with quasisymmetric lie in \(B_0\).


45P05 Integral operators
47B34 Kernel operators
Full Text: DOI MNR


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