Operator theory Invariant subspaces and dilations Lemma

Douglas' lemma

Builds on Hilbert space Hilbert space adjoint Positive operator Square root of a positive operator

Statement

Let $H_1$ , $H_2$ and $H$ be Hilbert spaces, and let $A \colon H_1 \to H$ and $B \colon H_2 \to H$ be bounded linear operators. The following are equivalent.

The range of $A$ is contained in the range of $B$ .
There is $\lambda \ge 0$ with $AA^* \le \lambda^2 BB^*$ , that is, $\norm{A^* x} \le \lambda \norm{B^* x}$ for all $x \in H$ .
There is a bounded linear operator $C \colon H_1 \to H_2$ with $A = BC$ .

Here $A^*$ and $B^*$ are the Hilbert space adjoints, and $\le$ is the order given by positive operators. Moreover, if these hold, there is a unique $C$ with $A = BC$ whose range lies in the closure of the range of $B^*$ . It satisfies $\ker C = \ker A$ and $\norm{C}^2 = \inf \set{\mu \ge 0 : AA^* \le \mu BB^*}$ .

Remarks

Proof idea: (3) implies (1) at once, and (3) implies (2) because $AA^* = BCC^*B^* \le \norm{C}^2 BB^*$ . For (2) implies (3), the map $B^* x \mapsto A^* x$ is well defined and bounded by $\lambda$ on the range of $B^*$ . Extend it to the closure, set it to $0$ on the orthogonal complement, and let $C$ be the adjoint of the result. For (1) implies (3), define $Cx$ as the unique $y \perp \ker B$ with $By = Ax$ , and use the closed graph theorem to see that $C$ is bounded.

Example: on $\ell^2$ let $B$ be the diagonal operator with entries $1/n$ and $A$ the diagonal operator with entries $1/n^2$ . Then $A = BC$ with $C$ diagonal with entries $1/n$ , and indeed $AA^* \le BB^*$ since $1/n^4 \le 1/n^2$ .

Going deeper

For a student

Douglas' lemma is a standard tool for factorization and for range inclusion questions. For example, with $B = (AA^*)^{1/2}$ , the square root of $AA^*$ , condition 2 holds in both directions, so $A$ and $(AA^*)^{1/2}$ have the same range. It is used in dilation theory, in the theory of de Branges spaces, and in the study of operator equations $A = BX$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Hilbert space concept
Hilbert space adjoint concept
Positive operator concept
Square root of a positive operator concept
Douglas' lemma statement

Tag	`NKTAN9XBZA`
URI	`m2://NKTAN9XBZA`
Kind	statement, lemma
Writing	W0 stub
MSC	47A05, 47A62
Status	This copy of the site has no build data. Run `m2 status NKTAN9XBZA`.
About	Hilbert space
Uses	Hilbert space, Hilbert space adjoint, Positive operator, Square root of a positive operator
Same as	Wikidata Q5301141 Wikipedia Douglas' lemma
Changed	2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

an inventory row: the row of the inventory that the node accounts for (0 of 1)
a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.