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.

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)