Give an example of a compact linear map on l2.
Consider the Hilbert space ( \ell^2 ) consisting of all square-summable sequences ( x = (x_1, x_2, x_3, \dots) ) with norm ( |x| = \left( \sum_{n=1}^\infty |x_n|^2 \right)^{1/2} ). Define a linear operator ( T: \ell^2 \to \ell^2 ) _______ ________ _________ _________ ___ __________.
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Let A be a normal operator on a Hilbert space X. Show that where
denotes the approximate eigen spectrum of A and
denotes the spectrum of A.
Let be a sequence of unitary operators in BL(H). Prove that if
, then A is unitary.
Define the spectral radius of a bounded linear operator . Find the spectral radius of A in
, where A is given by the matrix
with respect to the standard basis of .
Give one example of each of the following. Also justify your choice of example.
i) A self-adjoint operator on .
ii) A normal operator on a Hilbert space which is not unitary.
Give an example of a positive operator on
Consider the space C1[0,1] of all C1 functions on [0,1] endowed with the uniform norm induced from the space C[0,1], and consider the differential operator defined by
. Prove that D is linear, with closed graph, but not continuous. Can we conclude from here that C1[0,1] is not a Banach space? Justify your answer.
Read the proof of the closed graph theorem carefully and explain where and how we have used the following facts in the proof.
i) X is a Banach space.
ii) Y is a Banach space.
iii) F is a closed map.
iv) Which property of continuity is being established to conclude that F is continuous.