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:
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 with
. Give an example of a Cauchy sequence in X that do not converge in X. Justify your choice of example.
Let be a sequence of unitary operators in BL(H). Prove that if
, then A is unitary.
When is a normed linear space called separable? Show that a normed linear space is separable if its dual is separable [You should state all the proposition or theorems or corollaries used for proving the theorem]. Is the converse true? Give justification for your answer. [Whenever an example is given, you should justify that the example satisfies the requirements.]
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?
Consider the space c00. For , define
. Show that f is a linear functional which is not continuous w.r.t the norm
.