Give an example of a positive operator on
Consider the space ((\mathbb{C}^n, |\cdot|_2)). A linear operator (A: \mathbb{C}^n \to \mathbb{C}^n) is positive if it is self-adjoint and for all (x \in \mathbb{C}^n), ____ _______ _______ _______ ___ ___ ___ ____ _________.
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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?
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.
Let X be a normed space and Y be proper subspace of X. Show that the interior Y0 of Y is empty.
Let be a sequence of unitary operators in BL(H). Prove that if
, then A is unitary.
Prove the following result:
Let X be an inner product space and . Prove that
if and only if
.
Given an example of an Hilbert space H and an operator A on H such that is empty. Justify your choice of example.