Diagonal Linear Operator T in L(H): Examples

[Mikemaths]

I am trying to find an example of a diagonal linear operator T in L(H) H is hilbert space that is bounded but not compact and also one which is compact but not Hilbert-Schmidt. any Ideas??

Where diagonal means Ten=§en where § is the eigenvalue and en is on orthonormal basis.

 

[Landau]

(Instead of diagonal, I would say diagonalizable, since it depends on the chosen basis so is not an intrinsic property of the operator.)

The identity operator I is diagonalizable (well, I is of course diagonal w.r.t. every basis), bounded, but not compact.