One of the many irons in the fire that is my ongoing mathematical learning is a review of functional analysis motivated by my study of probability theory and stochastic processes. As I’ve mentioned before, I’ve been using the great text by Adam Bobrowski. In his discussion of discrete-time martingales, Bobrowski introduces a convergence theorem for monotone sequences of nonnegative (sometimes called positive semi-definite) operators. I would to like to give an exposition of the theorem, including a full proof. I would then like to give an application of this theorem, proving the existence of nonnegative square root for a nonnegative self-adjoint operator.
We say that a self-adjoint operator is nonnegative if for all ; we write . If are two self-adjoint operators such that , then we write .
Lemma 1. If is nonnegative, then is nonnegative for any .
Proof. If is even, then
Similarly, if is odd and , then
Lemma 2. (Nonnegative Operator Inequality) Let be a nonnegative, self-adjoint operator on a Hilbert space . Then for all .
Proof. If , then there is nothing to prove, so assume otherwise. Observe that proving the inequality above is equivalent to showing , where . Since is nonnegative, . Observe that is self-adjoint, being the difference of self-adjoint operators. By Cauchy-Schwarz,
Since , we see that
which completes the proof.
We now prove a convergence theorem for monotone sequences of self-adjoint operators.
Theorem 3. Suppose is a sequence of self-adjoint operators on a Hilbert space such that , for all , where is some constant. Then there exists a self-adjoint operator such that
In other words, converges to strongly (but not necessarily in the operator norm).
Proof. For any , the real sequence is nondecreasing and bounded from above by , hence converges to a real number . For any , the polarization identity lets us write
so the limit exists.
For any , we have by Cauchy-Schwarz that
so by Lemma 2, . By another application of Cauchy-Schwarz, we see that
Hence, for fixed, is a bounded complex-linear functional with respect to . By the Riesz representation theorem, there exists a unique element such that
is complex-linear, since
for all . Cauchy-Schwarz implies that , hence . is self-adjoint, since is self-adjoint for all and
for all . Being the difference of two self-adjoint operators, is self-adjoint and by Lemma 2,
This last quantity converges to as , for fixed, which completes the proof.
The preceding convergence theorem also applies to decreasing sequences of self-adjoint operators. If is a sequence of self-adjoint operators such that , for some constant , then the converge strongly to a self-adjoint operator . This result follows immediately from applying the preceding theorem to the sequence of operators .
We use the preceding convergence theorem for self-adjoint operators to prove the existence of a square root of self-adjoint operators .
Theorem 4. If is a self-adjoint operator such that for all , then there exists a self-adjoint operator such that , and commutes with all operators that commute with .
Proof. Without loss of generality, we may assume that . Otherwise, , in which case the theorem is obvious, or the operator satisfies and , where . Set , and observe that . We will find by an interative argument.
Define a sequence of operators inductively by and , for . The intuition behind this recursive inductive definition is that if exists (in the strong operator topology), then for any ,
Thus, is a positive square root of .
Induction shows that each is self-adjoint, nonnegative, and commutes with . Furthermore, induction shows that is polynomial in with positive coefficients and the observation
so that is polynomial in with positive coefficients. Hence, , for every . I claim that , for each . Clearly, . If , then self-adjoint implies that , so that . Hence,
By the monotone convergence theorem, there exists a strong limit . As shown above, , where , and since commutes with for each , taking the limit shows that , and therefore , commutes with as well.
We now derive some useful corollaries from Theorem 4, which have applications to the convergence of discrete-time martingales.
Corollary 5. Suppose is an increasing sequence of closed subspaces of a Hilbert space . Then the projections onto converge strongly to the projection onto the closed subspace .
If and , where is a filtration, then , where .
Proof. I first claim that . Indeed,
By the monotone convergence theorem for self-adjoint adjoint operators, converges strongly to an operator . We now show that is the projection operator .
Since , and therefore
we see that . Denote the range of by . Since , we see that
and therefore is closed. Since is self-adjoint, is the orthogonal projection onto the subspace . We need to show that . I claim that for each . Indeed, for and , , so , since the latter space is closed. By definition of the closure, . The reverse inclusion follows from observing that , where .
We know that since the pointwise limit of -measurable functions is -measurable. For the reverse inclusion, I claim that is a Dynkin system. Indeed, . If , then
If is a countable collection of pairwise disjoint sets, then by the dominated convergence theorem , where . , for each , so that . Since is closed under finite intersection, by Dynkin’s – lemma (see my notes), , which implies that .
Corollary 6. Let be a filtration of a probability space . For , the sequence defined by converges in -norm to , where .
Proof. Suppose . Then by Hölder’s inequality,
by Corollary 5. Since is dense in and conditional expectation is a Markov operator with norm , it follows that convergence holds on the entirety of (this is a standard -type argument, which I encourage the confused reader to work out).