TAILIEUCHUNG - Báo cáo toán học: "Quadratic interpolation "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: nội suy bậc hai. | Copyright by INCREST 1982 J. OPERATOR THEORY 7 1982 303-305 QUADRATIC INTERPOLATION p. R. HALMOS A long-known interpolation theorem 2 Theorem has recently played an important role 3 in the solution of the multiplicativity problem for Toeplitz operators 1 . The purpose of this note is to make that solution more accessible by offering a simple interpretation and proof of the pertinent special case. Neither interpolation theory nor Toeplitz operators are needed in what follows. The theorem has to do with what happens to an operator on an L2 space when the underlying measure is multiplied by a positive function. Suppose to be specific that Xis a measure space with measure Ịi and that w is a positive function on X. Positive means strictly positive here w x 0 for almost every X. All the functions that enter the discussion are required to be measurable. Both L2 dp and L2 wdp are Hilbert spaces. What sense does it make to say that an operator a bounded linear transformation acts on both these spaces One possible interpretation of such an expression in harmony with the point of view of classical analysis is that an operator on L2 dịi and an operator on L2 wd i agree on the elements common to their domains or equivalently if preferred agree on all functions that are finite linear combinations of characteristic functions of sets whose measure is finite in both senses . Suppose now that w0 and w are positive functions and that w is their geometric mean w2 WoWj . Suppose that an operator T acts on both Ho L2 w0dfiỴ and L2 w1d z . Does it then act on Hw L2 wdfi and if so what are the relations among the norms ỊT 0 II T lt and II T w on Ho Hỵ and Hw respectively The answer is imi i imnmii or in other words T W is dominated by the geometric mean of the norms T 0 and IIT t. This statement is the special case of the interpolation theorem the quadratic special case that was mentioned above and that is to be proved below. The theorem in exactly this form was called to my .

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