Diagonal operator
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.
![]() |
References
[1] | I.M. Singer, "Bases in Banach spaces" , 1 , Springer (1970) |
[2] | J. Wermer, "On invariant subspaces of normal operators" Proc. Amer. Math. Soc. , 3 : 2 (1952) pp. 270–277 |
[3] | I.D. Berg, "An extension of the Weyl–von Neumann theorem to normal operators" Trans. Amer. Math. Soc. , 160 (1971) pp. 365–371 |
Comments
For the notion of an unconditional basis see Basis.
For diagonal operators in the broad sense (and the corresponding notion of a diagonal algebra) see [a1].
References
[a1] | M. Takesaki, "Theory of operator algebras" , 1 , Springer (1979) pp. 259, 273 |
[a2] | P.R. Halmos, "A Hilbert space problem book" , Springer (1982) |
Diagonal operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonal_operator&oldid=46640