One of the most notorious open problems in functional analysis is the invariant subspace problem for Hilbert spaces, which I will state here as a conjecture:

Conjecture 1 (Invariant Subspace Problem, ISP0) Let ${H}$ be an infinite dimensional complex Hilbert space, and let ${T: H \rightarrow H}$ be a bounded linear operator. Then ${H}$ contains a proper closed invariant subspace ${V}$ (thus ${TV \subset V}$).

As stated this conjecture is quite infinitary in nature. Just for fun, I set myself the task of trying to find an equivalent reformulation of this conjecture that only involved finite-dimensional spaces and operators. This turned out to be somewhat difficult, but not entirely impossible, if one adopts a sufficiently generous version of “finitary” (cf. my discussion of how to finitise the infinitary pigeonhole principle). Unfortunately, the finitary formulation that I arrived at ended up being rather complicated (in particular, involving the concept of a “barrier”), and did not obviously suggest a path to resolving the conjecture; but it did at least provide some simpler finitary consequences of the conjecture which might be worth focusing on as subproblems.

I should point out that the arguments here are quite “soft” in nature and are not really addressing the heart of the invariant subspace problem; but I think it is still of interest to observe that this problem is not purely an infinitary problem, and does have some non-trivial finitary consequences.

I am indebted to Henry Towsner for many discussions on this topic.

— 1. Initial reductions —

The first reduction is to get rid of the closed invariant subspace ${V}$, as this will be the most difficult object to finitise. We rephrase ISP0 as

Conjecture 2 (Invariant Subspace Problem, ISP1) Let ${H}$ be an infinite dimensional complex Hilbert space, and let ${T: H \rightarrow H}$ be a bounded linear operator. Then there exist unit vectors ${v, w \in H}$ such that ${\langle T^n v, w \rangle = 0}$ for all natural numbers ${n \in {\bf N}}$.

Indeed, to see that ISP1 implies ISP0, we simply take ${V}$ to be the closed invariant subspace generated by the orbit ${v, Tv, T^2 v, \ldots}$, which is proper since it is orthogonal to ${w}$. To see that ISP0 implies ISP1, we let ${v}$ be an arbitrary unit vector in the invariant subspace ${V}$, and ${w}$ be an arbitrary unit vector in the orthogonal complement ${V^\perp}$.

The claim is obvious if ${H}$ is not separable (just let ${v}$ be arbitrary, and ${w}$ to be a normal vector to the separable space spanned by ${v, Tv, T^2 v, \ldots}$), so we may normalise ${H}$ to be ${\ell^2({\bf N})}$. We may also normalise ${T}$ to be a contraction (thus ${\|T\|_{op} \leq 1}$), and let ${(a_{ij})_{i,j \geq 1}}$ be the coefficients of ${T}$.

The next step is to restrict ${T}$ to a compact space of operators. Define a growth function to be a monotone increasing function ${F: {\bf N} \rightarrow {\bf N}}$. Given any growth function ${F}$, we say that a linear contraction ${T: \ell^2({\bf N}) \rightarrow \ell^2({\bf N})}$ with coefficients ${(a_{ij})_{i,j \geq 1}}$ is ${F}$-tight if one has the bound

$\displaystyle \sup_{1 \leq i \leq N} \sum_{j \geq F(N)} |a_{ij}|^2 \leq \frac{1}{N} \ \ \ \ \ (1)$

and

$\displaystyle \sup_{1 \leq j \leq N} \sum_{i \geq F(N)} |a_{ij}|^2 \leq \frac{1}{N}. \ \ \ \ \ (2)$

For instance, if the matrix ${(a_{ij})_{i,j \geq 1}}$ is band-limited to the region ${|j-i| \leq 10}$, it is ${F}$-tight with ${F(N) := N+11}$. If it is limited to the region ${i/2 \leq j \leq 2i}$, then it is ${F}$-tight with ${F(N) := 2N+1}$. So one can view ${F}$-tightness as a weak version of the band-limited property.

The significance of this concept lies in the following lemma:

Lemma 3 (Sequential compactness)

• (i) Every contraction ${T: \ell^2({\bf N}) \rightarrow \ell^2({\bf N})}$ is ${F}$-tight with respect to at least one growth function ${F}$.
• (ii) If ${F}$ is a growth function and ${T_1, T_2, \ldots}$ is a sequence of ${F}$-tight contractions, then there exists a subsequence ${T_{n_k}}$ which converges in the strong operator topology to an ${F}$-tight contraction ${T}$. Furthermore, the adjoints ${T_{n_k}^*}$ converge in the strong operator topology to ${T^*}$.

Proof: To prove (i), observe that if ${T}$ is a contraction and ${N \geq 1}$, then

$\displaystyle \sup_{1 \leq i \leq N} \sum_{j=1}^\infty |a_{ij}|^2 \leq 1$

and

$\displaystyle \sup_{1 \leq j \leq N} \sum_{i=1}^\infty |a_{ij}|^2 \leq 1$

and hence by the monotone convergence theorem we can find ${F(N)}$ such that (1), (2). By increasing ${F(N)}$ as necessary one can make ${F}$ monotone.

To prove (ii), we apply the usual Arzelá-Ascoli diagonalisation argument to extract a subsequence ${T_{n_k} = (a_{i,j,n_k})_{i,j \geq 1}}$ that converges componentwise (i.e. in the weak operator topology) to a limit ${T = (a_{i,j})_{i,j \geq 1}}$. From Fatou’s lemma we see that ${T}$ is an ${F}$-tight contraction. From the tightness one can upgrade the weak operator topology convergence to strong operator topology convergence (i.e.

$\displaystyle \lim_{k \rightarrow \infty} \sum_{j=1}^\infty |a_{i,j,n_k} - a_{i,j}|^2 = 0$

for all ${i}$) by standard arguments, and similarly for the adjoints. $\Box$

We will similarly need a way to compactify the unit vectors ${v, w}$. If ${F}$ is a growth function and ${0 < N_1 < N_2 < N_3 < \ldots}$ are natural numbers, we say that a unit vector ${v = (v_i)_{i=1}^\infty}$ is ${F,N_1,N_2,\ldots}$-tight if one has

$\displaystyle \sum_{i \geq N_k} |v_i|^2 \leq \frac{1}{F(N_{k-1})} \ \ \ \ \ (3)$

for all ${k \geq 1}$, with the convention that ${N_0=0}$. Similarly, we say that ${v}$ is ${F,N_1,\ldots,N_K}$-tight if one has (3) for all ${1 \leq k \leq K}$. One has the following variant of Lemma 3:

Lemma 4 (Sequential compactness) Let ${F}$ be a growth function.

• (i) Every unit vector ${v}$ is ${F,N_1,N_2,\ldots}$-tight with respect to at least one increasing sequence ${0 < N_1 < N_2 < \ldots}$. In fact any finite number of unit vectors ${v_1,\ldots,v_m}$ can be made ${F,N_1,N_2,\ldots}$-tight with the same increasing sequence ${0 < N_1 < N_2 < \ldots}$.
• (ii) If ${0 < N_1 < N_2 < \ldots}$, and for each ${k \geq 1}$, ${v_k}$ is a ${F,N_1,\ldots,N_k}$-tight unit vector, then there exists a subsequence ${v_{n_l}}$ of ${v_k}$ that converges strongly to an ${F,N_1,N_2,\ldots}$-tight unit vector ${v}$.

The proof of this lemma is routine and is omitted.

In view of these two lemmas, ISP0 or ISP1 is equivalent to

Conjecture 5 (Invariant Subspace Problem, ISP2) Let ${F}$ be a growth function, and let ${T = (a_{ij})_{i,j \geq 1}}$ be an ${F}$-tight contraction. Then there exist a sequence ${0 < N_1 < N_2 < \ldots}$ and a pair of ${F,N_1,N_2,\ldots}$-tight unit vectors ${v, w \in \ell^2({\bf N})}$ such that ${\langle T^n v, w \rangle = 0}$ for all natural numbers ${n \in {\bf N}}$.

The compactness given by the ${F}$-tightness and ${F,N_1,N_2,\ldots}$ will be useful for finitising later.

— 2. Finitising —

Now we need a more complicated object.

Definition 6 A barrier is a family ${{\mathcal T}}$ of finite tuples ${(N_1, \ldots, N_m)}$ of increasing natural numbers ${0 < N_1 < \ldots < N_m}$, such that

• (i) Every infinite sequence ${N_1 < N_2 < \ldots}$ of natural numbers has at least one initial segment ${(N_1,\ldots,N_m)}$ in ${{\mathcal T}}$; and
• (ii) If ${(N_1,\ldots,N_m)}$ is a sequence in ${{\mathcal T}}$, then no initial segment ${(N_1,\ldots,N_{m'})}$ with ${m' < m}$ lies in ${{\mathcal T}}$.

I learned the terminology “barrier” after asking this question on MathOverflow. Examples of barriers include

• The family of all tuples ${(N_1,\ldots,N_m)}$ of increasing natural numbers with ${m=10}$;
• The family of all tuples ${(N_1,\ldots,N_m)}$ of increasing natural numbers with ${m=N_1+1}$;
• The family of all tuples ${(N_1,\ldots,N_m)}$ of increasing natural numbers with ${m=N_{N_1+1}+1}$.

We now claim that ISP2 is equivalent to the following finitary statement. Let ${\ell^2(N) \equiv {\bf C}^N}$ denote the ${\ell^2}$ space on ${\{1,\ldots,N\}}$.

Conjecture 7 (Finitary invariant subspace problem, FISP0) Let ${F}$ be a growth function, and let ${{\mathcal T}}$ be a barrier. Then there exists a natural number ${N_*}$ such that for every ${F}$-tight contraction ${T: \ell^2(F(N_*)) \rightarrow \ell^2(F(N_*))}$, there exists a tuple ${(N_1,\ldots,N_m)}$ in ${{\mathcal T}}$ with ${0 < N_1 < \ldots < N_m < N_*}$, and ${F,N_1,\ldots,N_m}$-tight unit vectors ${v, w \in \ell^2(F(N_*))}$, such that ${|\langle T^n v, w \rangle| \leq \frac{1}{F(N_m)}}$ for all ${0 \leq n \leq F(N_m)}$.

We now show that ISP2 and FISP0 are equivalent.

Proof of ISP2 assuming FISP0. Let ${F}$ be a growth function, and let ${T}$ be an ${F}$-tight contraction. Let ${{\mathcal T}'}$ denote the set of all tuples ${0 < N_1 < \ldots < N_m}$ with ${m > 1}$ such that there does not exist ${F,N_1,\ldots,N_m}$-tight unit vectors ${v, w \in \ell^2({\bf N})}$ such that ${|\langle T^n v, w \rangle| \leq \frac{2}{m}}$ holds for all ${0 \leq n \leq m}$. Let ${{\mathcal T}}$ be those elements of ${{\mathcal T}'}$ that contain no proper initial segment in ${{\mathcal T}'}$.

Suppose first that ${{\mathcal T}}$ is not a barrier. Then there exists an infinite sequence ${0 < N_1 < N_2 < \ldots}$ such that ${(N_1,\ldots,N_m) \not \in {\mathcal T}}$ for all ${m}$, and thus ${(N_1,\ldots,N_m) \not \in {\mathcal T}'}$ for all ${m}$. In other words, for each ${m}$ there exists ${F,N_1,\ldots,N_m}$-tight unit vectors ${v_m,w_m \in \ell^2({\bf N})}$ such that ${|\langle T^n v_m, w_m \rangle| \leq \frac{2}{m}}$ for all ${0 \leq n \leq m}$. By Lemma 4, we can find a subsequence ${v_{m_j}, w_{m_j}}$ that converge strongly to ${F,N_1,N_2,\ldots}$-tight unit vectors ${v, w}$. We conclude that ${\langle T^n v, w \rangle=0}$ for all ${n \geq 0}$, and ISP2 follows.

Now suppose instead that ${{\mathcal T}}$ is a barrier. Let ${F'}$ be a growth function larger than ${F}$ to be chosen later. Then the ${F}$-tight contraction ${T}$ is also ${F'}$-tight, as is the restriction ${T|_N: \ell^2(N) \rightarrow \ell^2(N)}$ of ${T}$ to any finite subspace. Using FISP0, we can thus find ${0 < N_1 < \ldots < N_m < N_*}$ with ${(N_1,\ldots,N_m) \in {\mathcal T}}$ and ${F',N_1,\ldots,N_m}$-tight unit vectors ${v, w \in \ell^2(F'(N_*))}$ such that

$\displaystyle |\langle (T|_{F'(N_m)})^n v, w \rangle| \leq \frac{1}{F'(N_m)}$

for all ${0 \leq n \leq F'(N_m)}$, and in particular for all ${0 \leq n \leq m}$. Note that ${v, w}$ are almost in ${\ell^2(N_m)}$, up to an error of ${1/F'(N_{m-1})}$. From this and the ${F}$-tightness of the contraction ${T}$, we see (if ${F'}$ is sufficiently rapid) that ${(T|_{F'(N_m)})^n v}$ and ${T^n v}$ differ by at most ${1/m}$ for ${0 \leq n \leq m}$. We conclude that

$\displaystyle |\langle T^n v, w \rangle| \leq \frac{2}{m},$

and so ${(N_1,\ldots,N_m) \not \in {\mathcal T}}$, a contradiction. This yields the proof of ISP2 assuming FISP0.

Proof of FISP0 assuming ISP2. Suppose that FISP0 fails. Then there exists a growth function ${F}$ and a barrier ${{\mathcal T}}$ such that, for every ${N_*}$, there exists an ${F}$-tight contraction ${T_{N_*}: \ell^2(F(N_*)) \rightarrow \ell^2(F(N_*))}$ such that there does not exist any tuples ${(N_1,\ldots,N_m)}$ in ${{\mathcal T}}$ with ${0 < N_1 < \ldots < N_m < N_*}$, and ${F,N_1,\ldots,N_m}$-tight unit vectors ${v, w \in \ell^2(F(N_*))}$, such that ${|\langle T^n v, w \rangle| \leq \frac{1}{F(N_m)}}$ for all ${0 \leq n \leq F(N_m)}$.

We extend each ${T_{N_*}}$ by zero to an operator on ${\ell^2({\bf N})}$, which is still a ${F}$-tight contraction. Using Lemma 3, one can find a sequence ${N_{*,k}}$ going to infinity such that ${T_{N_{*,k}}}$ converges in the strong (and dual strong) operator topologies to an ${F}$-tight contraction ${T}$. Let ${F'}$ be a growth function larger than ${F}$ to be chosen later. Applying ISP2, there exists an infinite sequence ${0 < N_1 < N_2 < \ldots}$ and ${F',N_1,N_2,\ldots}$-tight unit vectors ${v, w \in \ell^2({\bf N})}$ such that ${\langle T^n v, w \rangle = 0}$ for all ${n \geq 0}$.

As ${{\mathcal T}}$ is a barrier, there exists a finite initial segment ${(N_1,\ldots,N_m)}$ of the above sequence that lies in ${{\mathcal T}}$. For ${k}$ sufficiently large, we have ${N_{*,k} \geq N_m}$, and also we see from the strong operator norm convergence of ${T_{N_{*,k}}}$ to ${T}$ (and thus ${T^n_{N_{*,k}}}$ to ${T^n}$ for any ${n}$, as all operators are uniformly bounded) that

$\displaystyle |\langle T^n_{N_{*,k}} v, w \rangle| \leq \frac{1}{F'(N_m)}$

for all ${0 \leq n \leq F(N_m)}$.

Now we restrict ${v, w}$ to ${\ell^2(F(N_{*,k}))}$, and then renormalise to create unit vectors ${v', w' \in \ell^2(F(N_{*,k}))}$. For ${k}$ large enough, we have

$\displaystyle \|v-v'\|, \|w-w'\| \leq 1 / F'(N_m)$

and we deduce (for ${F'}$ large enough) that ${v', w'}$ are ${F,N_1,N_2,\ldots,N_m}$-tight and ${|\langle T^n v, w \rangle| \leq \frac{1}{F(N_m)}}$ for all ${0 \leq n \leq F(N_m)}$. But this contradicts the construction of the ${T_{N_{*}}}$, and the claim follows.

— 3. A special case —

The simplest example of a barrier is the family of ${1}$-tuples ${(N)}$, and one of the simplest examples of an ${F}$-tight contraction is a contraction that is ${1}$-band-limited, i.e. the coefficients ${a_{ij}}$ vanish unless ${|i-j| \leq 1}$. We thus obtain

Conjecture 8 (Finitary invariant subspace problem, special case, FISP1) Let ${F}$ be a growth function and ${\epsilon > 0}$. Then there exists a natural number ${N_*}$ such that for every ${1}$-band-limited contraction ${T: \ell^2(F(N_*)) \rightarrow \ell^2(F(N_*))}$, there exists ${0 < N < N_*}$ and unit vectors ${v, w \in \ell^2(F(N_*))}$ with

$\displaystyle \sum_{j \geq N} |v_j|^2, \sum_{j \geq N} |w_j|^2 \leq \epsilon^2$

(i.e. ${v, w}$ are ${\epsilon}$-close to ${\ell^2(N)}$) such that ${|\langle T^n v, w \rangle| \leq \frac{1}{F(N)}}$ for all ${0 \leq n \leq F(N)}$.

This is perhaps the simplest case of ISP that I do not see how to resolve. (Note that the finite-dimensional operator ${T: \ell^2(F(N_*)) \rightarrow \ell^2(F(N_*))}$ will have plenty of (generalised) eigenvectors, but there is no particular reason why any of them are “tight” in the sense that they are ${\epsilon}$-close to ${\ell^2(N)}$.) Here is a slightly weaker version that I still cannot resolve:

Conjecture 9 (Finitary invariant subspace problem, special case, FISP2) Let ${F}$ be a growth function, let ${\epsilon > 0}$, and let ${T: \ell^2({\bf N}) \rightarrow \ell^2({\bf N})}$ be a ${1}$-band-limited contraction. Then there exists ${N > 0}$ and unit vectors ${v, w \in \ell^2({\bf N})}$ such that

$\displaystyle \sum_{j \geq N} |v_j|^2, \sum_{j \geq N} |w_j|^2 \leq \epsilon^2$

(i.e. ${v, w}$ are ${\epsilon}$-close to ${\ell^2(N)}$) such that ${|\langle T^n v, w \rangle| \leq \frac{1}{F(N)}}$ for all ${0 \leq n \leq F(N)}$.

This claim is implied by ISP but is significantly weaker than it. Informally, it is saying that one can find two reasonably localised vectors ${v, w}$, such that the orbit of ${v}$ is highly orthogonal to ${w}$ for a very long period of time, much longer than the degree to which ${v, w}$ are localised.