Block-diagonal operator

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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.


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



[a1] P.R. Halmos, "A Hilbert space problem book" , Springer (1982)
How to Cite This Entry:
Block-diagonal operator. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.K. Nikol'skiiB.S. Pavlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article