An operator
defined on the (closed) linear span of a
basis
in a normed (or only locally convex) space
by the equations
,
where
and where
are complex numbers. If
is a continuous operator, one has
If
is a Banach space, this condition is equivalent to the continuity of
if and only if
is an
unconditional basis
in
.
If
is an
orthonormal basis
in a Hilbert space
,
then
is a
normal operator,
and
,
while the spectrum of
coincides with the closure of the set
.
A normal and
completely-continuous operator
is a diagonal operator in the basis of its own
eigen vectors;
the restriction of a diagonal operator (even if it is normal) to its
invariant subspace
need not be a diagonal operator; given an
,
any normal operator
on a separable space
can be represented as
,
where
is a diagonal operator,
is a completely-continuous operator and
.
A diagonal operator in the broad sense of the word is an operator
of multiplication by a complex function
in the
direct integral of Hilbert spaces
i.e.
Cf.
Block-diagonal operator.