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# 7330hw2 - Math 7330 Functional Analysis Homework 2 Fall...

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Math 7330: Functional Analysis Fall 2002 Homework 2 A. Sengupta In the following, H is a complex Hilbert space with a Hermitian inner-product ( · , · ). All operators are operators on H . 1. Suppose P and Q are orthogonal projections. (i) Show that if PQ = QP then is an orthogonal projection. (ii) Show that, conversely, if is an orthogonal projection then = QP . 1

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2. Let P and Q be orthogonal projections. (i) Show that if PQ = P then = QP and Im( P ) Im( Q ). Show that the same conclusions hold if QP = P . (ii) Show that if Im( P ) Im( Q ) then QP = P . 2
3. Suppose A , B , C are mutually orthogonal closed subspaces of H , and let P A , P B , P C be the orthogonal projections onto A , B , C , respectively. Let X = A + B and Y = C + B , and let P X and P Y be the orthogonal projections onto X and Y , respectively. (i) Show that P X P Y = P Y P X . (ii) Express P X and P Y in terms of P A , P B and P C . (iii) Express P A , P B and P C in terms of P X and P Y . 3

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4. Suppose P and Q are orthogonal projections which commute, i.e. PQ = QP . The goal is to show that then the geometric situation of the preceding problem holds, i.e. there are mutually orthogonal closes subspaces A,B,C such that P is the orthogonal projection onto A + B and Q is the orthogonal projection onto C + B . Let R = PQ, S = P ( I - Q ) , T = Q ( I - P ) Observe that P = S + R and Q = T + R (i) Show that R , S , and T are orthogonal projections. [Note that if A is an orthogonal projection then so is I - A , and B commutes with A then it also commutes with I - A .] (ii) Show that RS = SR = 0, RT = TR = 0, and ST = TS = 0. (iii) Show that Im( R ), Im( S ), and Im( T ) are mutually orthogonal. Thus R , S , T are orthogonal projections onto mutually orthogonal closed subspaces. 4
5. Let x 1 ,x 2 3 ,... be a sequence of mutually orthogonal vectors in the Hilbert space H . Let S n = x 1 + ··· + x n . Let S 0 n = | x 1 | 2 + + | x n | 2 . (i) Show that for any integers m n , | S m - S n | 2 = S 0 m - S 0 n (ii) Show that the series n =1 x n to converge in H if and only if the series n | x n | 2 converges. 5

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Spectral Measures In the following, Ω is a non-empty set, B is a σ –algebra of subsets of Ω. A spectral measure is a mapping E from B to the set of all orthogonal projections on H satisfying the following conditions: (i) E ( ) = 0 (ii) E (Ω) = I (iii) if A 1 ,A 2 ,... ∈ B are mutually disjoint and their union is the set A then ( E ( A ) x,y ) = X n =1 ( E ( A n ) ) (1) for every H (iv) if A,B ∈ B then E ( A ) E ( B ) = E ( B ) E ( A ) = E ( A B ) For H deﬁne E x,y : B → C by E x,y ( A ) def = ( E ( A ) ) Conditions (i) and (iii) say that E x,y is a complex measure. If x = y we have E x,x ( A ) = ( E ( A ) x,x ) = | E ( A ) x | 2 0 (2) where we used the fact if P is any orthogonal projection then any x H decomposes as Px + x - with being perpendicular to x - and so ( Px,x ) = ( Px,Px + x - ) = ( ) + 0 = | | 2 (3) The non-negativity in (2) shows that E x,x is an (ordinary) measure on , B ) Recall that on the complex Hilbert space H any bounded linear operator A is determined uniquely by the “diagonal values” ( Ax,x ). It follows that if E and E 0 are spectral measures for which E x,x = E 0 x,x for all x H then E = E 0 .
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7330hw2 - Math 7330 Functional Analysis Homework 2 Fall...

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