tailieunhanh - Báo cáo toán học: "The invariant subspaces of a Volterra operator "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các subspaces bất biến của một nhà điều hành Volterra. | J. OPERATOR THEORY 6 1981 351-361 Copyright by INCREST 1981 THE PRODUCT OF SPECTRAL MEASURES w. RICKER. 1. INTRODUCTION The example of s. Kakutani 3 shows that the construction of the tensor product of two commuting spectral measures is not always possible. Accordingly if s and T are commuting scalar operators on a space X see 2 then s T and ST may not be of scalar type. It was pointed out by c. Foias for a discussion see 1 that the tensor product of two commuting spectral measures always exists if they are interpreted as spectral distributions. Then the product is of course only a spectral distribution and not necessarily a spectral measure. Accordingly the sum and the product of two commuting scalar operators are generalized scalar operators. As such they admit a functional calculus for smooth functions only. An alternative solution is possible if the operators s and T have extensions acting on a suitable larger space containing X. The technique of going to a larger space is often used in mathematical physics. For example the unbounded operator of differentiation in L2 R does not admit any eigenfunctions. However L2 R can be considered as part of a larger space which accommodates the complete set of eigenfunctions X - exp iẤx of the differentiation operator. Or let D be a self-adjoint second order non-singular differential operator with c coefficients on a b and let Du ý be an associated Sturm-Liouville problem with appropriate boundary conditions. Let u m ựộ be its solution for ự 6 C a Ố . Then m C a b - c a b is a Radon measure which does not have a density in the space C a b . However using the density of m with values in C a h 3 c a z the Green s function for the problem can be constructed see 9 . In this note we apply this time-honoured technique to construct the tensor product of two commuting spectral measures p and Q. Let p have domain Xể and Q have domain JA We shall seek a space Y containing X as a dense subspace such that for each EeXt and F e Ji the .