Give an example of a compact linear map on l2.
Consider the sequence space l2l^2, the space of square-summable sequences of real or complex numbers. Let T:l2→l2T: l^2 \to l^2 be a linear map defined ________ ________ _______ __________ ________ __________ ____ _________ ___ ______.
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Define a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-sehmidt operator a compact operator? Justify your answer.
Define the spectral radius of a bounded linear operator A ∈ BL(X). Find the spectral radius of A in BL, where A is given by the matrix
with respect to the standard basis of .
Let H=R³ and F be the set of all x = (x1, x2, x3) in H such that x1 = 0. Find F1. Verify that every x ∈ H can be expressed as x = y + z where y ∈ Fand z ∈ F1.
Let X = C00 with Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.
Prove the following result:
Suppose A is a non-zero compact self-adjoint operator on a Hilbert space H over K. Prove that there exists a finite set {r1,r2,..., rn} of a non-zero real numbers with and an orthonormal set {w1, w2,..., wn} in H such that
Further, mention in which step of the proof it is used that A is a compact self-adjoint operator. Explain why?
Given an example of an Hilbert space H and an operator A on Η such that σe(A)is empty. Justify your choice of example.
Let {An} be a sequence of unitary operators in BL(H). Prove that if , then A is unitary.