Diagonal operator

An operator $ D $ defined on the (closed) linear span of a basis $ \{ e _ {k} \} _ {k \geq 1 } $ in a normed (or only locally convex) space $ X $ by the equations $ De _ {k} = \lambda _ {k} e _ {k} $, where $ k \geq 1 $ and where $ \lambda _ {k} $ are complex numbers. If $ D $ is a continuous operator, one has

$$ \sup _ {k \geq 1 } | \lambda _ {k} | < + \infty . $$

If $ X $ is a Banach space, this condition is equivalent to the continuity of $ D $ if and only if $ \{ e _ {k} \} _ {k \geq 1 } $ is an unconditional basis in $ X $. If $ \{ e _ {k} \} _ {k \geq 1 } $ is an orthonormal basis in a Hilbert space $ H $, then $ D $ is a normal operator, and $ \| D \| = \sup _ {k \geq 1 } | \lambda _ {k} | $, while the spectrum of $ D $ coincides with the closure of the set $ \{ {\lambda _ {k} } : {k = 1 , 2 , . . . } \} $. A normal and completely-continuous operator $ N $ 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 $ \epsilon > 0 $, any normal operator $ N $ on a separable space $ H $ can be represented as $ N = D + C $, where $ D $ is a diagonal operator, $ C $ is a completely-continuous operator and $ \| C \| < \epsilon $.

A diagonal operator in the broad sense of the word is an operator $ D $ of multiplication by a complex function $ \lambda $ in the direct integral of Hilbert spaces

$$ H = \int\limits _ { M } \oplus H ( t) d \mu ( t) , $$

i.e.

$$ ( D f )( t) = \lambda ( t) f ( t) ,\ t \in M ,\ f \in H . $$

Cf. Block-diagonal operator.

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)