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.
i) A self-adjoint operator:
An example of a self-adjoint operator on a Hilbert space is the position operator x^\hat{x} in L2(R)L^2(\mathbb{R}), the space of square-integrable functions. The position operator x^\hat{x} acts on a function f(x)∈L2(R)f(x) \in L^2(\mathbb{R}) by x^f(x)=xf(x)\hat{x} f(x) = x f(x).
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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.]
Let f: C[0,1]→ be given by f (x) = x(1)∀x ∈ C[0,1]. Show that f is continuous w.r.t the supnorm and f is not continuous w.r.t the p-norm.
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 a Hilbert-Schmidt operator on a Hilbert space H and give an example. Is every Hilbert-sehmidt operator a compact operator? Justify your answer.
Which of the following maps are open? Give reasons for your answer.
i)
ii)
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.
Let {An} be a sequence of unitary operators in BL(H). Prove that if , then A is unitary.