block diagonalization of operator matrices


It's elementary linear algebra that every normal matrix is diagonalizable.
I'm
interested in the following possible generalization of this statement.

Let n be a positive integer, let H be a Hilbert space, and let N be a normal
element of M_n(B(H)), i.e., a normal operator on H^n.

Question: Is N block diagonalizable in the following sense? There are a
unitary scalar matrix u in M_n, and (necessarily normal) operators T_1, ...,
T_n in B(H) such that

N = (u \otimes 1)D(T_1, ..., T_n)(u^\ast \otimes 1),

where D(T_1, ..., T_n) denotes the block diagonal operator on H^n induced by
T_1, ..., T_n.

If such a block diagonalization does not always exists, then the next
question
would be about conditions forcing such a block diagonalization to exist.

Also, if H is infinite dimensional, we have a (non-canonical) isomorphism of
H
and H^n. This gives, for each operator in B(H) different matrix
representation. What if a block diagonalization exists with respect to one
such matrix representation - does then a block diagonalization exist with
respect to any matrix representation?

These questions don't seem to have a straightforward answer (at least not to
me), but they appear to be very natural, so that I'm sure that some work
must
have been done about them.

Any pertinent hints will be greatly appreciated!

Volker Runde.
.