Block-diagonal operator

with respect to a given orthogonal decomposition of a Hilbert space

A linear operator on which leaves each of the subspaces , , invariant. The spectrum of is the closure of the union of the spectra of the "blocks" , , . A block-diagonal operator in the broad sense of the word is an operator of multiplication by a function in the direct integral of Hilbert spaces

Here is a linear operator acting on the space . Each operator which commutes with a normal operator is a block-diagonal operator with respect to the spectral decomposition of this operator. See also Diagonal operator.

References

[1]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)

References

[a1]	P.R. Halmos, "A Hilbert space problem book" , Springer (1982)