Block-diagonal operator

with respect to a given orthogonal decomposition $ H = \sum _ {k \geq 1 } \oplus H _ {k} $ of a Hilbert space $ H $

A linear operator $ A $ on $ H $ which leaves each of the subspaces $ H _ {k} $, $ k \geq 1 $, invariant. The spectrum of $ A $ is the closure of the union of the spectra of the "blocks" $ A \mid _ {H _ {k} } = A _ {k} $, $ k \geq 1 $, $ \| A \| = \sup _ {k \geq 1 } \| A _ {k} \| $. A block-diagonal operator in the broad sense of the word is an operator $ A $ of multiplication by a function $ \lambda $ in the direct integral of Hilbert spaces

$$ H = \int\limits _ { M } \oplus H (t) d \mu (t) ,\ \ ( A f ) (t) = \lambda (t) f (t) ,\ t \in M . $$

Here $ \lambda (t) $ is a linear operator acting on the space $ H (t) $. 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)