# 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)