(b) Let H be a separable Hilbert space, and u₁, 2,... be a count-able orthonormal basis for H. Let (kn) be a bounded sequence ofscalars andA(x) = Σkn(x, un)un, TEH.Then it followsA*(x) = Σkn(x, Un) un, x=H.Since for all x EH,A* (A(x))A(A*(x)) ==72Σkn (A(x), Un)Un =72Σkn (A* (2), Un)Un =n|kn|²(x, Un) Un,|kn|²(x, un) un,| Requestexplainthesestepswe see that A is normal. Further, A is unitary if and only if |kn| = 1for each n and A is self-adjoint if and only if k₁ € R for each n.

Kindly refer to Step 2 for the solution.

Answered: A*(x) = n(t, Un)un, x≤ H. Since for all…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A*(x) = n(t, Un)un, x≤ H. Since for all x H, A* (A(x)) A(A*(x)) = Σkn(A(x), Un)Un = Σ|kn|²(x, Un)Un, Σkn (A*(x), Un)Un = Σ|kn|²(x, Un) un, 72 72 72 | Request explain these steps

A(x) = n(t, Un)un, x≤ H. Since for all x H, A (A(x)) A(A(x)) = Σkn(A(x), Un)Un = Σ|kn|²(x, Un)Un, Σkn (A(x), Un)Un = Σ|kn|²(x, Un) un, 72 72 72 | Request explain these steps