Characterising Weak-Operator

Continuous Linear Functionals on B(H) constructively

Douglas S. Bridges

Received: February 7, 2011

Communicated by Anil Nerode

Abstract. Let B(H) be the space of bounded operators on a notnecessarily-separable Hilbert space H. Working within Bishop-style constructive analysis, we prove that certain weak-operator continuous linear functionals on B(H) are finite sums of functionals of the form T ⇝ ⟨T x, y⟩. We also prove that the identification of weakand strong-operator continuous linear functionals on B(H) cannot be established constructively.

2010 Mathematics Subject Classification: 03F60, 47L50, 46S30 Keywords and Phrases: Constructive, operators, (ultra)weak operator topology, continuous functionals

1 Introduction

Let H be a complex Hilbert space that is nontrivial (that is, contains a unit vector), B(H) the space of all bounded operators on H, and B_{1}(H) the unit ball of B(H). In this paper we carry out, within Bishop-style constructive mathematics (BISH),^{1} an investigation of weak-operator continuous linear functionals on B(H). Depending on the context, we use, for example, x to represent either the ele-

ment (x_{1}, …, x_{N}) of the finite direct sum H_{N} ≡ ⨁_{n=1}^{N}H of N copies of H or else

the element (x_{n})_{n≥1} of the direct sum H_{∞} ≡ ⨁_{n≥1}H of a sequence of copies of

H. We use I to denote the identity projection on H. The following are the topologies of interest to us here.

1That is, mathematics that uses only intuitionistic logic and is based on a suitable set- or type-theoretic foundation [1, 2, 12]. For more on BISH see [3, 4, 8].

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▷ The weak operator topology: the weakest topology on B(H) with respect to which the mapping T ⇝ ⟨T x, y⟩ is continuous for all x, y ∈ H; sets of the form

{T ∈ B(H) : |⟨T x, y⟩| < ε} ,

with x, y ∈ H and ε > 0, form a sub-base of weak-operator neighbourhoods of 0 in B(H).

▷ The ultraweak operator topology: the weakest topology on B(H) with respect to which the mapping T ⇝ ∑_{n=1}^{∞} ⟨T x_{n}, y_{n}⟩ is continuous for all x, y ∈ H_{∞}; sets of the form

{T ∈ B(H) : |∑_{n=1}^{∞} ⟨T x_{n}, y_{n}⟩| < ε} ,

with x, y ∈ H_{∞} and ε > 0, form a sub-base of ultraweak-operator neighbourhoods of 0 in B(H).

These topologies are induced, respectively, by the seminorms of the form T ⇝ |⟨T x, y⟩| with x, y ∈ H, and those of the form T ⇝ |∑_{n=1}^{∞} ⟨T x_{n}, y_{n}⟩| with x, y ∈ H_{∞}. An important theorem in classical operator algebra theory states that the weak-operator continuous linear functionals on (any linear subspace of) B(H) all have the form T ⇝ ∑_{n=1}^{N} ⟨T x_{n}, y_{n}⟩ with x, y ∈ H_{N} for some N; and the ultraweak-operator continuous linear functionals on B(H) have the form T ⇝ ∑_{n=1}^{∞} ⟨T x_{n}, y_{n}⟩, where x, y ∈ H_{∞}. However, the classical proofs, such as those found in [10, 11, 14], depend on applications of nonconstructive versions of the Hahn-Banach theorem, the Riesz representation theorem, and polar decomposition. The foregoing characterisation of ultraweak-operator continuous functionals was derived constructively, when H is separable, in [9].^{2} A variant of it was derived in [8] (Proposition 5.4.16) without the requirement of separability, and using not the standard ultraweak operator topology but one that is classically, though not constructively, equivalent to it. Our aim in the present work is to provide a constructive proof of the standard classical characterisation of weakoperator continuous linear functionals (Theorem 10) on B(H), without the requirement of separability but with one hypothesis in addition to the classical ones. In presenting this work, we emphasise that, in contrast to their classical counterparts, our proofs contain extractable, implementable algorithms for the desired representation of weak-operator continuous linear functionals; moreover, the constructive proofs themselves verify that those algorithms meet their specifications.

2The characterisation was derived by Spitters in the case where H is separable and the subspace is B(H) itself ([15],Theorem 5); but his proof uses Brouwer’s continuity principle and so is intuitionistic, rather than in the style of Bishop.

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2 Preliminary Lemmas

The proof of our main theorem depends on a sequence of (at-times-complicated) lemmas. For the first one, we remind the reader of two elementary definitions in constructive analysis: we say that an inhabited set S—that is, one in which we can construct an element—is finitely enumerable if there exist a positive integer N and a mapping of {1, …, N} onto S; if that mapping is one-one, then S is called finite.

Lemma 1 If u is a weak-operator continuous linear functional on B(H), then there exist a finitely enumerable subset F of H × H and a positive number C such that |u(T)| ≤ C ∑_{(x,y)∈F} |⟨T x, y⟩| for all T ∈ B(H).

Proof. This is an immediate consequence of Proposition 5.4.1 in [8].

We shall need some information about locally convex spaces. Let (p_{j})_{j∈J} be a family of seminorms defining the topology on a locally convex linear space V, and let A be a subset of V. A subset S of A is said to be located (in A) if

inf {∑_{j∈F} p_{j}(x − s) : s ∈ S}

exists for each x ∈ A and each finitely enumerable subset F of J. We say that A is totally bounded if for each finitely enumerable subset F of J and each ε > 0, there exists a finitely enumerable subset S of A—called an εapproximation to S relative to (p_{j})_{j∈F} —such that for each x ∈ A there exists s ∈ S with ∑_{j∈F} p_{j}(x − s) < ε. The unit ball B_{1}(H) is weak-operator totally bounded ([8], Proposition 5.4.15); but, in contrast to the classical situation, it cannot be proved constructively that B_{1}(H) is weak-operator complete [5]. A mapping f between locally convex spaces (X, (p_{j})_{j∈J}) and (Y, (q_{k})_{k∈K}) is uniformly continuous on a subset S of X if for each ε > 0 and each finitely enumerable subset G of K, there exist δ > 0 and a finitely enumerable subset F of J such that if x, x′ ∈ S and ∑_{j∈F} p_{j}(x − x′) < δ, then ∑_{k∈G} q_{k} (f(x) − f(x′)) < ε. We recall four facts about total boundedness, locatedness, and uniform continuity in a locally convex space V. The proofs are found on pages 129–130 of [8].

▷ If f is a uniformly continuous mapping of a totally bounded subset A of V into a locally convex space, then f(A) is totally bounded.

▷ If f is a uniformly continuous, real-valued mapping on a totally bounded subset A of V, then sup_{x∈A} f(x) and inf_{x∈A} f(x) exist.

▷ A totally bounded subset of V is located in V.

▷ If A ⊂ V is totally bounded and S ⊂ A is located in A, then S is totally bounded.

We remind the reader that a bounded linear mapping T : X → Y between normed linear spaces is normed if its norm,

∥T ∥ ≡ sup {∥T x∥ : x ∈ X, ∥x∥ ≤ 1} ,

exists. If X is finite-dimensional, then ∥T ∥ exists; but the statement ‘Every bounded linear functional on an infinite-dimensional Hilbert space is normed’^{3} is essentially nonconstructive.

Lemma 2 Every weak-operator continuous linear functional on B(H) is normed.

Proof. This follows from observations made above, since, in view of Lemma 1, the linear functional is weak-operator uniformly continuous on the weakoperator totally bounded set B_{1}(H).

We note the following stronger form of Lemma 1.

Lemma 3 Let u be a weak-operator continuous linear functional on B(H).Then there exist δ > 0, and finitely many nonzero^{4} elements ξ_{1}, …, ξ_{N} and ζ_{1}, …, ζ_{N} of H with ∑_{n=1}^{N} ∥ξ_{n}∥^{2} = ∑_{n=1}^{N} ∥ζ_{n}∥^{2} = 1, such that |u(T)| ≤ δ ∑_{n=1}^{N} |⟨T ξ_{n}, ζ_{n}⟩| for each T ∈ B(H).

Proof. By Lemma 1, there exist a positive integer ν,C > 0, and vectors x, y ∈ H_{ν} such that |u(T)| ≤ C ∑_{n=1}^{ν} |⟨T x_{n}, y_{n}⟩| for all T ∈ B(H).^{5} For each n ≤ ν, construct nonzero vectors x′_{n}, y′_{n} such that x′_{n} ≠ x_{n} and y′_{n} ≠ y_{n}. The desired result follows from the inequality

∑_{n=1}^{ν} |⟨T x_{n}, y_{n}⟩| ≤ ∑_{n=1}^{ν} |⟨T (x_{n} − x′_{n}) , y_{n} − y′_{n}⟩| + ∑_{n=1}^{ν} |⟨T x′_{n}, y_{n} − y′_{n}⟩|

+ ∑_{n=1}^{ν} |⟨T (x_{n} − x′_{n}) , y′_{n}⟩| + ∑_{n=1}^{ν} |⟨T x′_{n}, y′_{n}⟩|

the fact that each of the vectors x′_{n}, x_{n} − x′_{n}, y′_{n}, and y_{n} − y′_{n} is nonzero, and scaling to get the desired norm sums equal to 1 and then the positive δ.

The next lemma will be used in an application of the separation theorem in the proof of Lemma 6.

3In fact, a nonzero linear functional on a normed space is normed if and only its kernel is located ([8], Proposition 2.3.6).

4A vector in a locally convex space is nonzero if it is mapped to a positive number by at least one seminorm.

5At this stage, it is trivial to prove Lemma 3 classically by simply deleting terms ⟨T x_{n}, y_{n}⟩ when either x_{n} or y_{n} is 0. With intuitionistic logic we need to work a little harder, because we cannot generally decide whether a given vector in H is, or is not, equal to 0.

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Lemma 4 Let ζ_{1}, …, ζ_{N} be elements of H with ∑_{n=1}^{N} ∥ζ_{n}∥^{2} = 1. Let K be a finite-dimensional subspace of H_{N}, and let ∥ ∥^{∗} be the standard norm on the dual space K^{∗} of K:

∥f∥^{∗} = sup {|f(x)| : x ∈ K, ∥x∥ ≤ 1} (f ∈ K^{∗}) .

Define a mapping F of B(H) into (K^{∗}, ∥ ∥^{∗}) by

F (T) (x) ≡ ∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩ (x ∈ K) .

Then F is weak-operator uniformly continuous on B_{1}(H).

Proof. Given ε > 0, let {x_{1}, …, x_{m}} be an ε-approximation to the (compact) unit ball of K. Writing x_{i} = (x_{i,1}, …, x_{i,N}), consider S, T ∈ B_{1}(H) with

∑_{i=1}^{m} ∑_{n=1}^{N} |⟨(S − T) x_{i,n}, ζ_{n}⟩| < ε.

For each x in the unit ball of K, there exists i such that ∥x − x_{i}∥ < ε. We compute

|F (S)(x) − F (T)(x)| ≤ |F (S)(x) − F (S)(x_{i})| + |F (S)(x_{i}) − F (T)(x_{i})|

+ |F (T)(x) − F (T)(x_{i})|

≤ ∑_{n=1}^{N} |⟨S (x_{n} − x_{i,n}) , ζ_{n}⟩| + ∑_{n=1}^{N} |⟨(S − T) x_{i,n}, ζ_{n}⟩|

+ ∑_{n=1}^{N} |⟨T (x_{n} − x_{i,n}) , ζ_{n}⟩|

≤ 2 ∑_{n=1}^{N} ∥x_{n} − x_{i,n}∥ ∥ζ_{n}∥ + ε

≤ 2 ∥x − x_{i}∥ ∥ζ∥ + ε < 3ε.

Hence ∥F (S) − F (T)∥^{∗} ≤ 3ε. Since ε > 0 is arbitrary, we conclude that F is uniformly continuous on B_{1}(H).

In order to ensure that the unit kernel B_{1}(H) ∩ ker u of a weak-operator continuous linear functional u on B(H) is weak-operator totally bounded, and hence weak-operator located, we derive a generalisation of Lemma 5.4.9 of [8].

Lemma 5 Let (V, (p_{j})_{j∈J}) be a locally convex space. Let V_{1} be a balanced, convex, and totally bounded subset of V. Let u be a linear functional on V that, on V_{1}, is both uniformly continuous and nonzero. Then V_{1} ∩ ker u is totally bounded.

Proof. Since u is nonzero and uniformly continuous on the totally bounded set V_{1},

C = sup{|u(y)| : y ∈ V_{1}}

exists and is positive. Choose y_{1} in V_{1} such that u(y_{1}) > C/2. Then

y_{0} ≡ \frac{C}{2u(y_{1})}y_{1}

belongs to the balanced set V_{1}, and u(y_{0}) = C/2. Let ε > 0, and let F be a finitely enumerable subset of J. Since each p_{j} is uniformly continuous on V, it maps the totally bounded set V_{1} onto a totally bounded subset of R.^{6} Hence there exists b > 0 such that p_{j}(x) ≤ b for each j ∈ F and each x ∈ V_{1}. Using Theorem 5.4.6 of [8], compute t with

0 < t < \frac{Cε}{C + 4b}

such that

S_{t} = {y ∈ V_{1} : |u(y)| ≤ t}

is totally bounded. Pick a t-approximation {s_{1}, …, s_{n}} of S_{t} relative to (p_{j})_{j∈F}, and set

y_{k} = \frac{C}{C + 2t}s_{k} − \frac{2}{C + 2t}u(s_{k})y_{0} (1 ≤ k ≤ n).

Then y_{k} ∈ ker(u). Since |u(s_{k})| ≤ t and V_{1} is balanced,

\frac{−u(s_{k})}{t}y_{0} ∈ V_{1}.

Thus

y_{k} = \frac{C}{C + 2t}s_{k} + (1 − \frac{C}{C + 2t}) (\frac{−u(s_{k})}{t}y_{0}) ∈ V_{1}.

6We use R and C for the sets of real and complex numbers, respectively.

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Now consider any element y of V_{1} ∩ ker(u). Since y ∈ S_{t}, there exists k such that ∑_{j∈F} p_{j}(y − s_{k}) < t and therefore

∑_{j∈F} p_{j}(y − y_{k}) ≤ ∑_{j∈F} p_{j}(y − s_{k}) + ∑_{j∈F} p_{j}(s_{k} − y_{k})

< t + \frac{2}{C + 2t} ∑_{j∈F} p_{j}(ts_{k} + u(s_{k})y_{0})

≤ t + \frac{2}{C + 2t} ∑_{j∈F} (tp_{j}(s_{k}) + u(s_{k})p_{j}(y_{0}))

≤ t + \frac{2t}{C} ∑_{j∈F} (p_{j}(s_{k}) + p_{j}(y_{0}))

≤ t (1 + \frac{4b}{C}) < ε.

Thus {y_{1}, …, y_{n}} is a finitely enumerable ε-approximation to V_{1} ∩ ker(u) relative to the family (p_{j})_{j∈F} of seminorms.

The next lemma, the most complicated in the paper, extracts much of the sting from the proof of our main theorem by showing how to find finitely many mappings of the form T ⇝ ⟨T x, ζ⟩ whose sum is small on the unit kernel of u.

Lemma 6 Let u be a nonzero weak-operator continuous linear functional on B(H). Let δ be a positive number, and ξ_{1}, …, ξ_{N} and ζ_{1}, …, ζ_{N} nonzero elements of H, such that^{7}

∑_{n=1}^{N} ∥ξ_{n}∥^{2} = ∑_{n=1}^{N} ∥ζ_{n}∥^{2} = 1,

and

|u(T)| ≤ δ ∑_{n=1}^{N} |⟨T ξ_{n}, ζ_{n}⟩| (T ∈ B(H)) .  (1)

Then for each ε > 0, there exists a unit vector x in the subspace

K ≡ Cξ_{1} × Cξ_{2} × ⋯ × Cξ_{N}

of H_{N}, such that x_{n} ≠ 0 for 1 ≤ n ≤ N and |∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩| < ε for all T ∈ B_{1}(H) ∩ ker u.

Proof. First note that since each ξ_{n} is nonzero, K is an N-dimensional subspace of H_{N}. Now, an application of Lemma 5 tells us that the unit kernel B_{1}(H) ∩ ker u of u is weak-operator totally bounded. For each x ∈ H_{N}, since the mapping T ⇝ ∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩ is weak-operator uniformly continuous on the unit kernel, we see that

7Such ξ_{k}, ζ_{k}, and δ exist, by Lemma 3.

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∥x∥_{0} = sup {|∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩| : T ∈ B_{1}(H) ∩ ker u}

exists. The mapping x ⇝ ∥x∥_{0} is a seminorm on H_{N} satisfying ∥x∥_{0} ≤ ∥ζ∥ ∥x∥ = ∥x∥; whence the identity mapping from (H_{N}, ∥ ∥) to (H_{N}, ∥ ∥_{0}) is uniformly continuous. Since the subset

{x ∈ K : ∥x∥ = 1}

of the finite-dimensional Banach space (K, ∥ ∥) is totally bounded, it follows that

β ≡ inf {∥x∥_{0} : x ∈ K, ∥x∥ = 1} ,

exists. It will suffice to prove that β = 0. For then, given ε with 0 < ε < 1, we can construct a unit vector x′ ∈ K such that |∑_{n=1}^{N} ⟨T x′_{n}, ζ_{n}⟩| < ε/2 for all T ∈ B_{1}(H) ∩ ker u. Picking nonzero vectors y_{n} ∈ Cξ_{n} such that

(∑_{n=1}^{N} ∥x′_{n} − y_{n}∥^{2})^{1/2} < ε/8, we have

|1 − (∑_{n=1}^{N} ∥y_{n}∥^{2})^{1/2}| < \frac{ε}{8},

so

x ≡ (∑_{n=1}^{N} ∥y_{n}∥^{2})^{−1/2} y

is a unit vector in Cξ_{1} × ⋯ × Cξ_{N} with each x_{n} ≠ 0. Moreover,

∥x − y∥^{2} = ∑_{k=1}^{N} |(∑_{n=1}^{N} ∥y_{n}∥^{2})^{−1/2} − 1|^{2} ∥y_{k}∥^{2}

≤ (\frac{ε}{8})^{2} ∑_{k=1}^{N} ∥y_{k}∥^{2}

≤ \frac{ε^{2}}{64} (1 + \frac{ε}{8})^{2} < \frac{ε^{2}}{16},

so for each T ∈ B_{1}(H) ∩ ker u,

|∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩| ≤ |∑_{n=1}^{N} ⟨T x′_{n}, ζ_{n}⟩| + ∑_{n=1}^{N} |⟨T (x_{n} − x′_{n}) , ζ_{n}⟩|

≤ \frac{ε}{2} + ∑_{n=1}^{N} ∥x_{n} − x′_{n}∥ ∥ζ_{n}∥

≤ \frac{ε}{2} + ∥x − x′∥ ∥ζ∥

≤ \frac{ε}{2} + ∥x − y∥ + ∥y − x′∥ < \frac{ε}{2} + \frac{ε}{4} + \frac{ε}{8} < ε.

To prove that β = 0, we suppose that β > 0. Then ∥ ∥_{0} is a norm equivalent to the original norm on K, so (K, ∥ ∥_{0}) is an N-dimensional Banach space. Define norms ∥ ∥^{∗} and ∥ ∥_{0}^{∗} on the dual K^{∗} of K by

∥f∥^{∗} ≡ sup {|f(x)| : x ∈ K, ∥x∥ ≤ 1} ,
∥f∥_{0}^{∗} ≡ sup {|f(x)| : x ∈ K, ∥x∥_{0} ≤ 1} .

For each T ∈ B(H) and each x ∈ K let

F (T) (x) ≡ ∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩ .

Then, by Lemma 4, F is weak-operator uniformly continuous as a mapping of B_{1}(H) into (K^{∗}, ∥ ∥^{∗}); since the norms ∥ ∥^{∗} and ∥ ∥_{0}^{∗} are equivalent on the finite-dimensional dual space K^{∗}, F is therefore weak-operator uniformly continuous as a mapping of B_{1}(H) into (K^{∗}, ∥ ∥_{0}^{∗}). Hence

D = F (B_{1}(H) ∩ ker u)

is a totally bounded, and therefore located, subset of (K^{∗}, ∥ ∥_{0}^{∗}). Moreover, for each T ∈ B_{1}(H) ∩ ker u and each x ∈ K, |F (T) (x)| ≤ ∥x∥_{0}; so D is a subset of the unit ball S_{0}^{∗} of (K^{∗}, ∥ ∥_{0}^{∗}). We shall use the separation theorem from functional analysis to prove that D is ∥ ∥_{0}^{∗}-dense in S_{0}^{∗}. Consider any ϕ in S_{0}^{∗}, and suppose that

0 < d = inf {∥ϕ − F (T)∥_{0}^{∗} : T ∈ B_{1}(H) ∩ ker u} .

Now, D is bounded, convex, balanced, and located; so, by Corollary 5.2.10 of [8], there exists a linear functional v on (K^{∗}, ∥ ∥_{0}^{∗}) with norm 1 such that

v(ϕ) > |v(F (T))| + \frac{d}{2} (T ∈ B_{1}(H) ∩ ker u) .

It is a simple exercise^{8} to show that since (K^{∗}, ∥ ∥_{0}^{∗}) is N-dimensional, there exists y ∈ K such that ∥y∥_{0} = 1 and v(f) = f(y) for each f ∈ K^{∗}. Hence

ϕ(y) ≥ sup \{|F (T)(y)| : T ∈ B_{1}(H) ∩ ker u\} + \frac{d}{2}

> sup {|∑_{n=1}^{N} ⟨T y_{n}, ζ_{n}⟩| : T ∈ B_{1}(H) ∩ ker u} = ∥y∥_{0} ,

which contradicts the fact that ϕ ∈ S_{0}^{∗}. We conclude that d = 0 and therefore that D is ∥ ∥_{0}^{∗}-dense in S_{0}^{∗}. Continuing our proof that β = 0, pick T_{0} ∈ B_{1}(H) with u(T_{0}) > 0. Replacing u by u(T_{0})^{−1}u if necessary, we may assume that u(T_{0}) = 1. Define a linear functional Ψ on (K, ∥ ∥_{0}) by setting

Ψ(x) = β ∑_{n=1}^{N} ⟨T_{0}x_{n}, ζ_{n}⟩ (x ∈ K) .

Note that for x ∈ K we have

|Ψ(x)| ≤ β ∑_{n=1}^{N} ∥x_{n}∥ ∥ζ_{n}∥ ≤ β ∥x∥ ∥ζ∥ ≤ ∥x∥_{0.}

Hence Ψ ∈ S_{0}^{∗}. By the work of the previous paragraph, we can find T ∈ B_{1}(H)∩ ker u such that ∥Ψ − F (T)∥_{0}^{∗} < β/2δ. In particular, since ∥ξ∥_{0} ≤ ∥ξ∥ = 1,

|∑_{n=1}^{N} ⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩| < \frac{β}{2δ}.  (2)

In order to apply the defining property of δ and thereby obtain a contradiction, we need to estimate not the sum on the left hand side of (2), but ∑_{n=1}^{N} |⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩|. To do so, we write

{n : 1 ≤ n ≤ N} = P ∪ Q,

where P, Q are disjoint sets,

n ∈ P ⇒ ⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩ ≠ 0, and

n ∈ Q ⇒ |⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩| < \frac{β}{2δN} .

If n ∈ P, we set

λ_{n} = \frac{1}{⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩} |⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩| ,

8Alternatively, we can refer to [3] (page 287, Theorem 10) or [8] (Theorem 5.4.14).

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and if n ∈ Q, we set λ_{n} = 0; in each case, we define γ_{n} ≡ λ_{n}ξ_{n}. Then γ ≡ (γ_{1}, …, γ_{N}) ∈ K and

∥γ∥_{0}^{2} ≤ ∥γ∥^{2} = ∑_{n=1}^{N} |λ_{n}|^{2} ∥ξ_{n}∥^{2} ≤ ∥ξ∥^{2} = 1.

Hence

|∑_{n=1}^{N} ⟨(βT_{0} − T) γ_{n}, ζ_{n}⟩| ≤ ∥Ψ − F (T)∥_{0}^{∗} < \frac{β}{2δ} .

Moreover,

|∑_{n=1}^{N} ⟨(βT_{0} − T) γ_{n}, ζ_{n}⟩| = |∑_{n∈P} ⟨(βT_{0} − T) λ_{n}ξ_{n}, ζ_{n}⟩|

= ∑_{n∈P} |⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩| ,

so

∑_{n=1}^{N} |⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩|

= ∑_{n∈P} |⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩| + ∑_{n∈Q} |⟨(βT_{0} − T) ξ_{n}, ζ_{n}⟩|

≤ |∑_{n=1}^{N} ⟨(βT_{0} − T) γ_{n}, ζ_{n}⟩| + N (\frac{β}{2δN} ) < \frac{β}{δ}

and therefore u(βT_{0} − T) < β. But u(βT_{0} − T) = βu(T_{0}) − u(T) = β, a contradiction which ensures that β actually equals 0.

We shall apply Lemma 6 shortly; but its application requires another construction.

Lemma 7 Let N be a positive integer, let ξ_{1}, …, ξ_{N} be linearly independent vectors in H, and let ζ_{1}, …, ζ_{N} be nonzero elements of H, such that ∑_{n=1}^{N} ∥ξ_{n}∥^{2} = ∑_{n=1}^{N} ∥ζ_{n}∥^{2} = 1. Then there exists a positive number c with the following property: for each unit vector z in the subspace

K ≡ Cξ_{1} × ⋯ × Cξ_{N},

there exists T ∈ B_{1}(H) such that ∑_{n=1}^{N} ⟨T z_{n}, ζ_{n}⟩ > c.

Proof. Let

m ≡ inf {∥ζ_{n}∥^{2} : 1 ≤ n ≤ N} > 0.

Define a norm on the N-dimensional span V of {ξ_{1}, …, ξ_{N}} by

∥∑_{n=1}^{N} α_{n}ξ_{n}∥_{1} ≡ max_{1≤n≤N} |α_{n}| .

Since V is finite-dimensional, there exists b > 0 such that ∥x∥_{1} ≤ b ∥x∥ for each x ∈ V. Let z ≡ (λ_{1}ξ_{1}, …, λ_{N}ξ_{N}) in H_{N} satisfy ∥z∥ = 1. If |λ_{n}| < 1/\sqrt{N} for each n, then

1 = ∑_{n=1}^{N} ∥λ_{n}ξ_{n}∥^{2} = ∑_{n=1}^{N} |λ_{n}|^{2} ∥ξ_{n}∥^{2} < ∑_{n=1}^{N} (\frac{1}{\sqrt{N}} )^{2} = 1,

which is absurd. Hence we can pick ν such that |λ_{ν}| > 1/\sqrt{2N}. Define a linear mapping T on H such that

T ξ_{ν} = \frac{λ_{ν}^{∗}}{b |λ_{ν}|}ζ_{ν}, T ξ_{n} = 0 (n ≠ ν) ,

and T x = 0 whenever x is orthogonal to V. Then

∥T (∑_{n=1}^{N} α_{n}ξ_{n})∥ = \frac{|λ_{ν}^{∗}|}{b |λ_{ν}|} |α_{ν}| ≤ \frac{1}{b} ∥∑_{n=1}^{N} α_{n}ξ_{n}∥_{1} ≤ ∥∑_{n=1}^{N} α_{n}ξ_{n}∥ ,

so T ∈ B_{1}(H). Moreover,

⟨T z_{n}, ζ_{n}⟩ = \{ \begin{matrix}0 & if n ≠ ν \\ \frac{1}{b} |λ_{ν}| ∥ζ_{ν}∥^{2} & if n = ν, \end{matrix}

so

∑_{n=1}^{N} ⟨T z_{n}, ζ_{n}⟩ = \frac{1}{b} |λ_{ν}| ∥ζ_{ν}∥^{2} > \frac{m}{b\sqrt{2N}} .

It remains to take c ≡ m/b\sqrt{2N}.

The next lemma takes the information arising from the preceding two, and shows that when the vectors ξ_{n} in (1) are linearly independent, we can approximate u by a finite sum of mappings of the form T ⇝ ⟨T x, y⟩, not just on its unit kernel but on the entire unit ball of B(H). At the same time, we produce a priori bounds on the sums of squares of the norms of the components of the vectors x, y that appear in the terms ⟨T x, y⟩ whose sum approximates u(T). Those bounds will be needed in the proof of our characterisation theorem.

Lemma 8 Let H be a Hilbert space, and u a nonzero weak-operator continuous linear functional on B(H). Let δ be a positive number, ξ_{1}, …, ξ_{N} linearly independent vectors in H, and ζ_{1}, ⋯, ζ_{N} nonzero vectors in H, such that

∑_{n=1}^{N} ∥ξ_{n}∥^{2} = ∑_{n=1}^{N} ∥ζ_{n}∥^{2} = 1

and (1) holds. Let c > 0 be as in Lemma 7.Then for each ε > 0, there exists x ∈ Cξ_{1} × ⋯ × Cξ_{N} such that x_{n} ≠ 0 for each n,

∥x∥ < \frac{2 ∥u∥}{c},

and

|u(T) − ∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩| < ε

for all T ∈ B_{1}(H).

Proof. Pick T_{0} ∈ B_{1}(H) with u(T_{0}) > 0. To begin with, take the case where u(T_{0}) = 1 and therefore ∥u∥ ≥ 1. Given ε > 0, set

α ≡ \frac{min \{ε, 1\}}{2 ∥u∥ (1 + ∥u∥)}.

Applying Lemma 6, we obtain nonzero vectors z_{n} ∈ Cξ_{n} (1 ≤ n ≤ N) such that ∑_{n=1}^{N} ∥z_{n}∥^{2} = 1 and

|∑_{n=1}^{N} ⟨T z_{n}, ζ_{n}⟩| < cα (T ∈ B_{1}(H) ∩ ker u) .

For each T ∈ B_{1}(H), since

(1 + ∥u∥)^{−1} (T − u(T)T_{0}) ∈ B_{1}(H) ∩ ker u,

we have

|∑_{n=1}^{N} ⟨(T − u(T)T_{0}) z_{n}, ζ_{n}⟩| < (1 + ∥u∥) cα.

By Lemma 7, there exists T_{1} ∈ B_{1}(H) such that ∑_{n=1}^{N} ⟨T_{1}z_{n}, ζ_{n}⟩ > c. We compute

c < ∑_{n=1}^{N} ⟨T_{1}z_{n}, ζ_{n}⟩

≤ |∑_{n=1}^{N} ⟨(T_{1} − u(T_{1})T_{0}) z_{n}, ζ_{n}⟩| + |u(T_{1})| |∑_{n=1}^{N} ⟨T_{0}z_{n}, ζ_{n}⟩|

≤ (1 + ∥u∥)cα + ∥u∥ |∑_{n=1}^{N} ⟨T_{0}z_{n}, ζ_{n}⟩| .

Hence

|∑_{n=1}^{N} ⟨T_{0}z_{n}, ζ_{n}⟩| > \frac{c}{∥u∥} (1 − (1 + ∥u∥) α)

≥ \frac{c}{∥u∥} (1 − \frac{1}{2 ∥u∥}) > \frac{c}{2 ∥u∥},

since ∥u∥ ≥ 1. Setting

x ≡ (∑_{n=1}^{N} ⟨T_{0}z_{n}, ζ_{n}⟩)^{−1} z,

we have 0 ≠ x_{n} ∈ Cξ_{n} for each n, and

∥x∥ = |∑_{n=1}^{N} ⟨T_{0}z_{n}, ζ_{n}⟩|^{−1} ∥z∥ < \frac{2 ∥u∥}{c}.

Moreover, for each T ∈ B_{1}(H),

|u(T) − ∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩| = |∑_{n=1}^{N} ⟨T_{0}z_{n}, ζ_{n}⟩|^{−1} |∑_{n=1}^{N} ⟨(u(T)T_{0} − T) z_{n}, ζ_{n}⟩|

< \frac{2 ∥u∥}{c}(1 + ∥u∥)cα ≤ ε.

We now remove the restriction that u(T_{0}) = 1. Applying the first part of the theorem to v ≡ u(T_{0})^{−1}u, we construct y ∈ K such that each component y_{n} ≠ 0, ∥y∥ ≤ 2 ∥v∥ /c, and

|v(T) − ∑_{n=1}^{N} ⟨T y_{n}, ζ_{n}⟩| < u(T_{0})^{−1}ε,

and we obtain the desired conclusion by taking x ≡ u(T_{0})y.

Lemma 9 Under the hypotheses of Lemma 8, but without the assumption that u is nonzero, for all ε, ε′ > 0, there exists x ∈ Cξ_{1} ×⋯×Cξ_{N} such that x_{n} ≠ 0 for each n,

∥x∥ < \frac{2 (∥u∥ + ε′)}{c},

and

|u(T) − ∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩| < ε

for all T ∈ B_{1}(H).

Proof. Either ∥u∥ > 0 and we can apply Lemma 8, or else ∥u∥ < ε/2. In the latter event, pick x in Cξ_{1} × ⋯ × Cξ_{N} such that x_{n} ≠ 0 for each n and

∥x∥ < min \{\frac{ε}{2}, \frac{2 (∥u∥ + ε′)}{c}\} .

Then for each T ∈ B_{1}(H) we have

|∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩| ≤ ∑_{n=1}^{N} ∥x_{n}∥ ∥ζ_{n}∥ ≤ ∥x∥ ∥ζ∥ < \frac{ε}{2}

and therefore

|u(T) − ∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩| ≤ ∥u∥ + |∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩| < ε.

3 The Characterisation Theorem

We are finally able to prove our main result, by inductively applying Lemma 9.

Theorem 10 Let H be a nontrivial Hilbert space, and u a nonzero weakoperator continuous linear functional on B(H). Let δ be a positive number, ξ_{1}, …, ξ_{N} linearly independent vectors in H,^{9} and ζ_{1}, ⋯, ζ_{N} nonzero vectors in H, such that |u(T)| ≤ δ ∑_{n=1}^{N} |⟨T ξ_{n}, ζ_{n}⟩| for all T ∈ B(H). Then there exists x ∈ Cξ_{1} × ⋯ × Cξ_{N} such that

u(T) = ∑_{n=1}^{N} ⟨T x_{n}, ζ_{n}⟩  (3)

for all T ∈ B(H).

Proof. Re-scaling if necessary, we may assume that ∥u∥ < 2^{−3}. In the notation of, and using, Lemma 9, compute x^{(1)} in K ≡ Cξ_{1} × ⋯ × Cξ_{N} such that^{10}

∥x^{(1)}∥ ≤ \frac{2}{c} (∥u∥ + 2^{−3}) < \frac{1}{2c}

and

|u(T) − ∑_{n=1}^{N} ⟨T x_{n}^{(1)}, ζ_{n}⟩| < 2^{−4} (T ∈ B_{1}(H)) .

Suppose that for some positive integer k we have constructed vectors x^{(i)} ∈ K (1 ≤ i ≤ k) such that

∥x^{(k)}∥ < \frac{1}{2^{k}c},  (4)

u(T) = ∑_{n∈F} ⟨T ξ_{n}, λ_{n}ζ_{n}⟩ ,

9The requirement that the vectors ξ_{n} be linearly independent is the one place where we have a stronger hypothesis than is needed in the classical theorem. It is worth noting here that if u(T) has the desired form ∑_{n=1}^{N} ⟨T ξ_{n}, ζ_{n}⟩, then classically we can find a set F of indices n ≤ N such that (i) the set S of those ξ_{n} with n ∈ F is linearly independent and (ii)

if ξ_{k} ∉ S, then ξ_{k} is linearly dependent on S. We can then write with each λ_{n} ∈ C. Constructively, this is not possible, since we cannot necessarily determine whether or not ξ_{n} is linearly dependent on the vectors ξ_{1}, …, ξ_{n−1}.

10In this proof we do not need the fact that, according to Lemma 9, we can arrange for the components of the vector x^{(1)}, and of the subsequently constructed vectors x^{(k)}, to be nonzero.

Documenta Mathematica 16 (2011) 597–617

and

|u(T) − ∑_{n=1}^{N} ⟨T (x_{n}^{(1)} + ⋯ + x_{n}^{(k)}) , ζ_{n}⟩| < 2^{−k−3} (T ∈ B_{1}(H)) . (5)

Consider the weak-operator continuous linear functional

v : T ⇝ u(T) − ∑_{n=1}^{N} ⟨T (x_{n}^{(1)} + ⋯ + x_{n}^{(k)}) , ζ_{n}⟩

on B(H). Writing

x_{n}^{(1)} + ⋯ + x_{n}^{(k)} = λ_{n}ξ_{n}

and

γ ≡ max {|λ_{1}| , …, |λ_{n}|} ,

for each T ∈ B(H) we have

|v(T)| ≤ |u(T)| + ∑_{n=1}^{N} |⟨T (x_{n}^{(1)} + ⋯ + x_{n}^{(k)}) , ζ_{n}⟩|

≤ δ ∑_{n=1}^{N} |⟨T ξ_{n}, ζ_{n}⟩| + ∑_{n=1}^{N} |λ_{n}| |⟨T ξ_{n}, ζ_{n}⟩|

≤ (δ + γ) ∑_{n=1}^{N} |⟨T ξ_{n}, ζ_{n}⟩| .

We can now apply Lemma 9, to obtain

x^{(k+1)} = (x_{1}^{(k+1)}, …, x_{N}^{(k+1)}) ∈ K

such that

∥x^{(k+1)}∥ < \frac{2}{c} (∥ν∥ + 2^{−k−3}) < \frac{1}{2^{k+1}c}

and

|u(T) − ∑_{n=1}^{N} ⟨T (x_{n}^{(1)} + ⋯ + x_{n}^{(k)} + x_{n}^{(k+1)}) , ζ_{n}⟩|

= |v(T) − ∑_{n=1}^{N} ⟨T x_{n}^{(k+1)}, ζ_{n}⟩| < 2^{−k−4}

for all T ∈ B_{1}(H). This completes the inductive construction of a sequence (x^{(k)})_{k≥1} in K such that (4) and (5) hold for each k. The series ∑_{k=1}^{∞} x^{(k)} converges to a sum x in the finite-dimensional Banach space K, by comparison with ∑_{k=1}^{∞} 2^{−k}c^{−1}. Letting k → ∞ in (5), we obtain (3) for all T ∈ B_{1}(H) and hence for all T ∈ B(H).

For nonzero u, the proof of our theorem can be simplified at each stage of the induction, since we can use Lemma 8 directly. If H has dimension > N, we can then construct the classical representation of u in the general case as follows. Either ∥u∥ > 0 and there is nothing to prove, or else ∥u∥ < δ (the same δ as in the statement of the theorem). In the latter case, we construct a unit vector ξ_{N+1} orthogonal to each of the vectors ξ_{n} (1 ≤ n ≤ N), set ζ_{N+1} = ξ_{N+1}, and consider the weak-operator continuous linear functional

v : T ⇝ u(T) + δ ⟨T ξ_{N+1}, ζ_{N+1}⟩ .

We have

|v(T)| ≤ |u(T)| + δ |⟨T ξ_{N+1}, ζ_{N+1}⟩| ≤ δ ∑_{n=1}^{N+1} |⟨T ξ_{n}, ζ_{n}⟩| .

Moreover,

|v(I)| ≥ δ ∥ξ_{N+1}∥^{2} − |u(I)| ≥ δ − ∥u∥ > 0,

where I is the identity operator on H; so v is nonzero. We can therefore apply the nonzero case to v, to produce a vector y ∈ Cξ_{1} × ⋯ × Cξ_{N+1} such that

v(T) = ∑_{n=1}^{N+1} ⟨T y_{n}, ζ_{n}⟩ (T ∈ B(H)) .

Setting x_{n} = y_{n} (1 ≤ n ≤ N) and x_{n+1} = y_{N+1} − δξ_{N+1}, we obtain

u(T) = ∑_{n=1}^{N+1} ⟨T x_{n}, ζ_{n}⟩

for each T ∈ B(H). Note, however, that this proof gives x in Cξ_{1} ×⋯×Cξ_{N} × Cξ_{N+1}, not, as in Theorem 10, in Cξ_{1} × ⋯ × Cξ_{N}. As an immediate consequence of Theorem 10, the functional u therein is a linear combination of the functionals T ⇝ ⟨T ξ_{n}, ζ_{n}⟩ associated with the seminorms that describe the boundedness of u:

Corollary 11 Under the hypotheses of Theorem 10, there exist complex numbers α_{1}, …, α_{N} such that

u(T) = ∑_{n=1}^{N} α_{n} ⟨T ξ_{n}, ζ_{n}⟩

for each T ∈ B(H).

4 Strong-operator Continuous Functionals

Next we turn briefly to the strong operator topology on B(H): the locally convex topology generated by the seminorms T ⇝ ∥T x∥ with x ∈ H. (That is, the weakest topology with respect to which the mapping T ⇝ T x is continuous for each x ∈ H.) Clearly, a weak-operator continuous linear functional on B(H) is strong-operator continuous. The converse holds classically, but, as we now show by a Brouwerian example, is essentially nonconstructive. Let (e_{n})_{n≥1} be an orthonormal basis of unit vectors in an infinite-dimensional Hilbert space, and let (a_{n})_{n≥1} be a binary sequence with at most one term equal to 1. Then for k ≥ j we have

∑_{n=j}^{k} |a_{n} ⟨T e_{1}, e_{n}⟩| ≤ (∑_{n=j}^{k} a_{n}^{2})^{1/2} (∑_{n=j}^{k} |⟨T e_{1}, e_{n}⟩|^{2})^{1/2}

≤ (∑_{n=j}^{k} |⟨T e_{1}, e_{n}⟩|^{2})^{1/2} .

Since ∑_{n=1}^{∞} |⟨T e_{1}, e_{n}⟩|^{2} converges to ∥T e_{1}∥, we see that ∑_{n=j}^{k} |a_{n} ⟨T e_{1}, e_{n}⟩| → 0 as j, k → ∞. Hence

u(T) ≡ ∑_{n=1}^{∞} a_{n} ⟨T e_{1}, e_{n}⟩

defines a linear functional u on B(H); moreover, |u(T)| ≤ ∥T e_{1}∥ for each T, so (by Proposition 5.4.1 of [8]) u is strong-operator continuous. Suppose it is also weak-operator continuous. Then, by Lemma 2, it is normed. Either ∥u∥ < 1 or ∥u∥ > 0. In the first case, if there exists (a unique) ν with a_{ν} = 1, then u(T) = ⟨T e_{1}, e_{ν}⟩ for each T ∈ B(H). Defining T such that T e_{1} = e_{ν} and T e_{n} = 0 for all n ≠ ν, we see that T ∈ B_{1}(H) and u(T) = 1; whence ∥u∥ = 1, a contradiction. Thus in this case, a_{n} = 0 for all n. On the other hand, in the case ∥u∥ > 0 we can find T such that u(T) > 0; whence there exists n such that a_{n} = 1. It now follows that the statement

If H is an infinite-dimensional Hilbert space, then every strongoperator continuous linear functional on B(H) is weak-operator continuous

implies the essentially nonconstructive principle

LPO: For each binary sequence (a_{n})_{n≥1}, either a_{n} = 0 for all n or else there exists n such that a_{n} = 1

and so is itself essentially nonconstructive.

5 Concluding Observations

The ideal constructive form of Theorem 10 would have two improvements over the current one. First, the requirement that the vectors ξ_{n} be linearly independent would be relaxed to have them only nonzero in Lemma 8, Lemma 9, and Theorem 10. Second, B(H) would be replaced by a suitable linear subspace R of itself, and our theorem would apply to linear functionals that are weak-operator continuous on R, where “suitable” probably means “having weak-operator totally bounded unit ball R_{1} ≡ R ∩ B_{1}(H)”. With that notion of suitability and with minor adaptations, Lemma 6 holds and the proof of Lemma 8 goes through as far as the construction of the vector z ∈ K. In fact, Theorem 10 goes through with B(H) replaced by any linear subspace R of B(H) that has weak-operator totally bounded unit ball and satisfies the following condition (cf. Lemma 7):

(*) Let N be a positive integer, let ξ_{1}, …, ξ_{N} be linearly independent vectors in H, and let ζ_{1}, …, ζ_{N} be nonzero elements of H, such that ∑_{n=1}^{N} ∥ξ_{n}∥^{2} = ∑_{n=1}^{N} ∥ζ_{n}∥^{2} = 1. Then there exists a positive number c with the following property: for each unit vector z in the subspace

K ≡ Cξ_{1} × ⋯ × Cξ_{N}

there exists T ∈ R_{1} such that ∑_{n=1}^{N} ⟨T z_{n}, ζ_{n}⟩ > c.

This condition holds in the special case where N = 1, in which case, if also R_{1} is weak-operator totally bounded, we obtain Theorem 1 of [6].^{11} However, there seems to be no means of establishing (*) for N > 1 and a general R. So the ideal form of our theorem remains an ideal and a challenge.

Acknowledgements. This work was supported by a Marie Curie IRSES award from the European Union for the project Construmath, with counterpart funding from the New Zealand Ministry of Research, Science & Technology. It was written while the author enjoyed the hospitality of (i) Professor Helmut Schwichtenberg and his colleagues in the Logic Group at the Mathematisches Institut der Ludwig-Maximilians-Universität München, and (ii) Professor Erik Palmgren in the Matematiska Institution, University of Uppsala. The author also thanks the referee for providing helpful comments that improved the presentation of the paper.

11But the proof of the theorem in [6] is simpler and more direct than the case N = 1 of the proof of our Theorem 10 above.

Documenta Mathematica 16 (2011) 597–617

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Douglas S. Bridges Department of Mathematics & Statistics University of Canterbury Private Bag 4800 Christchurch 8140 New Zealand d.bridges@math.canterbury.ac.nz 618

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