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 .
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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.
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.
Let X be a Banach space and Y be a closed subspace of X. Let be canonical quotient map. Show that
is open.
Let X be an inner product space and . Prove that
if and only if
.
et and F be the set of all
in H such that
. Find
. Verify that every
can be expressed as
where
and
Let be given by
. Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.
Let X, Y be normed spaces and suppose BL(X, Y) and CL(X, Y) denote, respectively, the space of bounded linear operators from X to Y and the space of compact linear operators from X to Y. Show that CL(X, Y) is linear subspace of BL(X, Y). Also, Show that if Y is a Banach space, then CL(X, Y) is a closed subspace of BL(X, Y).