Prove the following result:
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 of a non-zero real numbers with
and an orthonormal set
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?
Let (A) be a non-zero compact self-adjoint operator on a Hilbert space (H) over (\mathbb{K}). We want to show the existence of a finite set of non-zero real numbers ({r_1, r_2, \dots, r_n}) with (|r_1| \ge |r_2| \ge \dots \ge |r_n|) and an orthonormal set ({w_1, w_2, \dots, w_n}) in (H) such that [ A(x) = \sum_{i=1}^n r_i \langle x, w_i\rangle w_i, \quad x \in H. ] --- Step 1: Spectral theorem for compact self-adjoint operators _____ _____ ______ ____ ___ ____ ___ ______ __________ _____.
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Prove the following result:
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 of a non-zero real numbers with
and an orthonormal set
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?
State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
for
is a norm.
b) C0 is a Banach space.
c) If A is the right shift operator on l2, then the eigen spectrum is non-empty.
d) If a normed linear space is reflexive, then so is its dual space.
e) If a normed linear space X is finite dimensional, then so is X'.
Let be a sequence of unitary operators in BL(H). Prove that if
, then A is unitary.
Let X be an inner product space and . Prove that
if and only if
.
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.
et and F be the set of all
in H such that
. Find
. Verify that every
can be expressed as
where
and
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).