Give an example of a positive operator on .
A positive operator \( T \) on \( \mathbb{C}^n \) with the Euclidean norm \( \| \cdot \|_2 \) is an operator that satisfies the condition: \[ \langle __________ ____ _____ _______ ________ ______ ___.
____ _________ ______ _______ ___ _____ _______ ___ __________ ______.
____ _______ ______ _________ ______ ____ ________ _____ ________ ______ _____ __________.
_________ __________ _____ ___ _____ _____ ___ __________ ______ _________ ________.
_______ ____ ________ ______ _______ ______ ________ ______ ___ ________ ______ ___.
__________ ________ ____ __________ ___.
_________ _______ _______ ________ _____ ________ _______.
____ _______ ___ ____ ______.
_____ ____ ___ ___ _____ ___ _______.
____ _____ _______ _______ ____ _______ _________.
___ ______ _______ _________ ________ __________ _______.
__________ _________ __________ ________ _____ ______.
___ _______ __________ _______ ________ _______ ____ __________ ________ ___ ____ ____.
_____ _____ _____ _________ __________ __________ _____ ______ ___ ____.
_______ _____ _____ __________ ___ _______ ________ _____ ______ ___.
____ ________ __________ ___ ____ _________ _______ ______ _____ ______.
____ __________ _________ ________ ___ ___ ________.
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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.
Let A be a normal operator on a Hilbert space X. Show that σ(A) ⊂ σa(A) where σa (A) denotes the approximate eigen spectrum of A and σ(A) denotes the spectrum of A.
State whether the following statements True or False? Justify your answers:
a) The function defined on
as:
b) Co 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'.
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 .
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 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.
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.