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MATH 611 (Spring 2009) Homework #8 Due April 16th (1) (a) Let T ∈ B(H1 , H2 ) and S ∈ B(H2 , H3 ) where the Hj are Hilbert spaces. Prove that (ST )∗ = T ∗ S ∗ . (b) Let T ∈ B(H, K) be invertible with a bounded inverse. Prove that T ∗ is invertible with a bounded inverse and that (T ∗ )−1 = (T −1 )∗ . (2) Let T ∈ B(H, K). Prove that T is an isometric isomorphism if and only if T is a unitary equivalence. (3) Let {un }∞ n=1 be an orthonormal basis for a Hilbert space H. Prove that for all x, y ∈ H (x, y) = ∞ X (x, un )(un , y). n=1 (4) Let f ∈ L∞ and let Tf ∈ B(L2 ) be the multiplication operator defined by Tf (g) = f · g. Prove that (Tf )∗ = Tf . (5) An operator is said to have rank 1 if its range is one-dimensional. Let T ∈ B(H, K) be a bounded rank 1 operator between Hilbert spaces and let ψ be a non-zero vector in the range of T . Show that there exists a φ ∈ H so that T (x) = (x, φ)ψ for all x ∈ H and that kT k = kφk kψk. Find a formula for T ∗ in terms of φ and ψ. Additional Questions (Not to be handed in) (6) Prove that T ∗∗ = T and kT ∗ k = kT k. (7) Prove that en (x) = exp(inx) for n ∈ Z is an orthonormal set in L2C ([−π, π]). (8) Prove that any separable infinite dimensional Hilbert spaces is unitarily equivalent to `2 .