Answer to Question #4653 in Functional Analysis for sharmistha saha

Question

1)Give one example of a normal operator on a hilbert space which is not unitary.Justify.
2)Give one example of a normed space which is not an inner product.Justify.

Answer 1

1) By definition a normal operator on a complex Hilbert space H is a continuous
linear operator
N:H --> H
that commutes with its hermitian adjoint
N*:
N N* = N* N.

A unitary operator is an operator U:H --> H
satisfying the identity
& U U* = U* U = I.

Therefore if U is any
unitary operator, e.g the identity U=I, and
a =/= 1 is an arbitrary complex
number distinct from 1, then&
& N=aU
is normal.
Indeed,& N* = aU*,
whence
N N* = (a U) (a U*) = a^2 (U U*) = a^2 (U* U) = (a U*) (a U) = N*
N.

2) Fix p>=1 and consider the space l^p consisting of all infinite
sequences
x = (x1, x2, ... )
such that
sum_{i=1}^{infinity}
|xi|^p < infinity
Then l^p is a hilbert space only for p=2.
For all
other p it is a normes space but not an inner space.


Actually, for a
norm |*| on a linear space V to be induced by inner product <*,*>, so
|x|^2 = <x,x>
it is necessary and sufficient that
|x+y|^2 +
|x-y|^2 = 2(|x|^2 + |y|^2)
for any x,y from V.