# Block-diagonal operator

From Encyclopedia of Mathematics

*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) |

#### Comments

#### References

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

**How to Cite This Entry:**

Block-diagonal operator. N.K. Nikol'skiiB.S. Pavlov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Block-diagonal_operator&oldid=13450

This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098