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)