- #1

Karl Karlsson

- 104

- 12

- Homework Statement
- Let ##V = \mathbb{C}([0, 1], \mathbb{C})## be the vector space of continuous functions ##f: [0,1] \rightarrow \mathbb{C}## with inner product:

$$\langle f|g \rangle = \int_0^1 \overline {f(t)}g(t)\;\mathrm{d}t$$

For ##h \in V## , let ##L_h: V \rightarrow V## be the operator defined by ##L_h(f)(t)=h(t)f(t)##.

b) Determine the adjoint operator to ##L_h##

c) For which functions is the operator ##L_h## self adjoint?

d) For which functions is the operator ##L_h## unitary?

e) Give example of one function h(t) such that the operator ##L_h## is self adjoint and has at least two different eigenvalues

- Relevant Equations
- $$\langle f|g \rangle = \int_0^1 \overline {f(t)}g(t)\;\mathrm{d}t$$

##L_h(f)(t)=h(t)f(t)##

b)

c and d):

In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct?

e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find twtherwise does b-d above look correct?

Thanks in advance!

c and d):

In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct?

e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find twtherwise does b-d above look correct?

Thanks in advance!