Description

Linear Operators
Assignment 6
Exercise 6.1
Let x ∈ ℓ2, i.e., x = (xn) is a square-summable sequence. Define the operator L: ℓ2 → ℓ2 by Lx = y,
, n = 1,2,3,…
where the coefficients anm, n,m ∈ N {0} satisfy ∞ . Any such L is called a Hilbert-
Schmidt operator on ℓ2.
i) Verify that y ∈ ℓ2, i.e., that L is well-defined. (2Marks)
ii) Find the matrix elements Lij := ⟨ei,Lej⟩ of L with respect to the standard basis {en}n∈N, with en := (0,…,0,1,0,…), where the 1 is in the nth position. (1Mark)
iii) Show that L is bounded with ∥L∥ ≤ M. (2Marks) iv) Show that L defined by

is a Hilbert-Schmidt operator. Find the operator norm ∥L∥. What can you say about its relationship to
M2?
(3Marks)