Brownian Motion and Martingales in Analysis (The Wadsworth by Richard Durrett

By Richard Durrett

This publication could be of curiosity to scholars of arithmetic.

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Extra resources for Brownian Motion and Martingales in Analysis (The Wadsworth Mathematics Series)

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Ud, k = 1, 2, ... in H and associate to T an m x m-matrix where m = dim H, l~m~oo: (6) Consider the Fourier expansions of xlEI Hand Tx with respect to {Uk}: x = Lk akuko Tx = Lk {3kUk' (7) Then the sequence {13k} can be expressed in terms of the {aJ as {3j = L tjkako k Vj. (8) Indeed, {3j = (Tx, u) = (L aku k ko T*Uj) = L k ak(u k , T*uj ) = . L tjkak· k This calculation provides the convergence of the series in (8). The matrix (6) gives the matrix representation of T in the basis {Uk}' The matrix of an operator in general 38 CHAPTER 2 depends on the choice of a basis.

Theorem 6 states that each of the three objects <1>, T, S uniquely determines the other two. Suppose we are given an operator T. Then is defined by the first and S by the second of the formulae in (8). The functional is called a sesqui-linear form of the operator T, (x) is called the quadratic form of T, S is called the adjoint operator of T. The adjoint operator is denoted by T *. It is defined, in accordance with (8), by (Tx, y) = (x, T*y), V x, Y E H. (12) Theorem 6 ensures the existence and uniqueness of the adjoint operator.

Suppose we are given an operator T. Then is defined by the first and S by the second of the formulae in (8). The functional is called a sesqui-linear form of the operator T, (x) is called the quadratic form of T, S is called the adjoint operator of T. The adjoint operator is denoted by T *. It is defined, in accordance with (8), by (Tx, y) = (x, T*y), V x, Y E H. (12) Theorem 6 ensures the existence and uniqueness of the adjoint operator. Equality (11) means I Til = I (13) T* II· It is clear from (12) that T** := (T*)* = T, (aiTI + azTz)* = air:, + azTi, (TITz)* = TiT~.

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