Abstract
Abstract. For a sequence (A j ) of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums Sn = n j=1 A j in a "strong" sense. Those limits turn out to be some special idempotent operators (unbounded, in general). In the case of X = L 2 (Ω, µ), an arbitrary unbounded closed and densely defined operator A in X may be the µ-almost sure limit of Sn (i.e. Snf → Af µ-a.e. for all f ∈ D(A)).
Abstract
Introduction. Monotone families of projections are important objects in both classical and functional analysis. Let us mention here the huge classical theory of Fourier series with respect to general or special orthonormal systems of functions, the theory of martingales, the spectral theory of normal operators in a Hilbert space or, more generally, the theory of spectral or well-bounded operators in a Banach space ([3], [5] ). In the case of non-selfadjoint projections, the assumption that the systems of idempotent operators considered are uniformly bounded is important and, as a rule, necessary if we want to reach results similar to the classical well-known ones for selfadjoint projections in a Hilbert space. The most typical results of this kind are the integral spectral representations for well-bounded or power-bounded operators ([1], [11], [12]).
Abstract
The main goal of this paper is to consider some "unbounded situations". We discuss the convergence problems concerning series of mutually orthogonal projections in Banach spaces. We do not assume that the partial sums of those series are bounded in the operator norm. This implies, in particular, that every unbounded closed and densely defined operator A in L 2 (µ)

Introduction. Monotone families of projections are important objects in both classical and functional analysis. Let us mention here the huge classical theory of Fourier series with respect to general or special orthonormal systems of functions, the theory of martingales, the spectral theory of normal operators in a Hilbert space or, more generally, the theory of spectral or well-bounded operators in a Banach space ( [3] , [5] ). In the case of non-selfadjoint projections, the assumption that the systems of idempotent operators considered are uniformly bounded is important and, as a rule, necessary if we want to reach results similar to the classical well-known ones for selfadjoint projections in a Hilbert space. The most typical results of this kind are the integral spectral representations for well-bounded or power-bounded operators ( [1] , [11] , [12] ).

The main goal of this paper is to consider some "unbounded situations". We discuss the convergence problems concerning series of mutually orthogonal projections in Banach spaces. We do not assume that the partial sums of those series are bounded in the operator norm. This implies, in particular, that every unbounded closed and densely defined operator A in L 2 (µ) is the sum, in the sense of the almost sure convergence, of a certain series of mutually orthogonal idempotent operators. It should be stressed here that the possibility of almost sure approximation of A by the multiples of orthogonal projections in L 2 (µ) depends heavily on the properties of the spectral measure of |A| (cf. [7] , [8] ). 0. Generalities. Let X be a Banach space. For any fixed pair (F, K) of linear subspaces of X (not necessarily closed) satisfying the condition F ∩ K = (Θ), we can define an operator A by putting (1) D(A) = F ⊕ K and A(f + k) = f for f ∈ F, k ∈ K.

Obviously, Ax = A 2 x for x ∈ D(A). Such an A will be called an idempotent operator . A may not be closed or densely defined. Such a general definition is motivated by the situations which will be described later and which appear in natural circumstances.

In the sequel, we shall often write A = (F, K) or A = (F A , K A ). A bounded idempotent defined on the whole space X is called a projection in X. Then, obviously, F and K are closed subspaces of X and X = F ⊕ K. Let us remark that an idempotent A = (F, K) is closable iff F ∩ K = (Θ). Indeed, let A be the closure of A; then Ax = x, Ay = Θ for any x ∈ F , y ∈ K. Thus x = Ax = Θ for any x ∈ F ∩ K. On the other hand, if F ∩ K = (Θ), then for any f n ∈ F , k n ∈ K with f n + k n → x and A(f n + k n ) → f , we have f n → f ∈ F and k n → k ∈ K. This means that (F , K) represents the closure (F, K) − of (F, K). Clearly, the idempotent (F, K) is closed iff the subspaces F and K are closed.

1. Quasi-strong convergence of orthogonal series of projections in a Banach space 1.1. Projections A i , i ∈ I, are said to be mutually orthogonal iff A i A j = 0 for i = j, which is equivalent to F i ⊆ K j for i = j. For two projections A and B, we write A ≤ B iff AB = BA = A. Evidently, if the projections

. . are mutually orthogonal projections. It is easy to check that if A 1 , A 2 , . . . are mutually orthogonal projections and A j = (F j , K j ), then

Obviously, A ≤ B iff F A ⊆ F B and K A ⊇ K B (for more details, see e.g. [4] ).
Let
A j = (F j , K j ) (j = 1, 2, . .
.) be mutually orthogonal projections

Convergence of orthogonal series of projections
S n x exists} and let
Convergence of orthogonal series of projections
Let us put
Convergence of orthogonal series of projections
, there exist a sequence (f 1 , f 2 , . . .), f j ∈ F j , and k n ∈ n j=1 K j (n = 1, 2, . . .), uniquely determined by x and such that
Convergence of orthogonal series of projections
In particular, we have f j = A j x. Thus x ∈ D(A) iff f 1 + . . . + f n strongly converges to Ax as n → ∞ and k n → k ∈ K. Consequently, (4) leads us to the equality
Convergence of orthogonal series of projections
The idempotent S = (F 0 , K) satisfying (2), (3) (not necessarily bounded or densely defined) will be called a quasi-strong limit of S n = n j=1 A j . We shall also write q.s.-lim n→∞ S n = S or q.s.-
Convergence of orthogonal series of projections
Obviously, in the case when the projections A j act in a Hilbert space and are selfadjoint, then S is an orthogonal projection (onto F 0 = F 0 ). In general (i.e. when A j are not necessarily selfadjoint), the operator S can be unbounded. It may be closed and densely defined but also it may happen that it is not closed or densely defined. It may also not be closable, even if X is a Hilbert space.
Convergence of orthogonal series of projections
In the sequel, we shall use the following notation. For vectors x, y, . . . in a Banach space X, the symbol [x, y, . . .] denotes the closed subspace of X spanned by x, y, . . . We shall also write [F i ; i ∈ I] = [ i∈I F i ] for a family (F i ) i∈I of subspaces. If X is a Hilbert space, [x, y, . . .]
Convergence of orthogonal series of projections
∧ will stand for the orthogonal projection onto [x, y, . . .], and, for e ∈ X, we will write e = [e] ∧ = ·, e e/ e 2 .
1.3.
Example (S is unbounded, densely defined and closed). Let H be a separable Hilbert space and let (e 1 , e 2 , . . . , f 1 , f 2 , . . .) be an orthonormal basis in H. We put
1.3.
For the sequence S n = (F n , K n ), S 0 = 0, we have S n ≤ S n+1 , which means that S n = n i=1 A i with mutually orthogonal projections
1.3.
Then the assumption that ϕ = f + k, with f ∈ F , k ∈ K, leads directly to a contradiction. Thus the idempotent (F, K) is unbounded, densely defined and closed. It remains to prove that S n → S = (F, K) quasi-strongly, i.e. D(S) = {x ∈ H : S n x converges strongly} = F ⊕ K, and
1.3.
To do this, let us remark that, for x ∈ H, we have the unique representation
1.3.
We shall first show that, for such an x (of the form (5)),
1.3.
and, for x of the form (5),
1.3.
On the other hand,
1.3.
By the uniqueness of coefficients for x of the form (5), we have
1.3.
R e m a r k. For any idempotent ( F H, KH) in a Hilbert space H with F , K orthogonal projections, and for any finite-dimensional orthogonal projections P 1 ≤ P 2 ≤ . . . tending to identity and commuting with F and K, the projections ( P n F H, (
1.3.
tend quasi-strongly to ( F H, KH). 
Example (S is defined only on an arbitrary closed infinite
-dimen- sional subspace F ). Let (f is , g s ; i, s = 1, 2, . . .) be a basis in H with F = [f i,s ; i, s = 1, 2, . . .]. Put A i = ∞ s=1 ·, f is + g s f is . Then, obviously, A i are mutually orthogonal projections. Moreover, n i=1 A i f js = f js for n > j, and n i=1 A i x = n i=1 ∞ s=1 f is x → x for x ∈ F as well as n i=1 A i g s = n i=1 f is and n i=1 A i x 2 = n x 2 → ∞ for x ∈ [g s ; s = 1, 2, , . . .].
Example (S is densely defined but not closable). Let (e

and define
Indeed, suppose that f n ∈ K n , where, obviously,
and define
. . , e n , f n+1 , e n+3 , e n+4 , . . .] for n = 0, 1, . . .
and define
c s e s .
R. Jajte and A. Paszkiewicz
Consequently, 0 = f n , e n+2 = α f n+1 , e n+2 = α sin 1 2 n+1 , so α = 0 and
R. Jajte and A. Paszkiewicz
which is impossible.
R. Jajte and A. Paszkiewicz
Obviously,
R. Jajte and A. Paszkiewicz
We know that S = (F 0 , K), where
R. Jajte and A. Paszkiewicz
the series is strongly convergent
R. Jajte and A. Paszkiewicz
1.6. Example (S is not closed but closable with the closure 1). Let, as before, (e 1 , e 2 , . . .) be an orthonormal basis in H and let 
R. Jajte and A. Paszkiewicz
Then, as in Example 1.5, K n is of the form (7) and (F n , K n ) denotes a projection for any n = 0, 1, . . .
It remains to prove that
This is equivalent to x = Θ.
It remains to prove that
We are going to characterize those densely defined idempotents which are quasi-strong sums of series of mutually orthogonal projections. To clarify the situation a little bit, let us first consider the case (F 0 , (Θ)) with F 0 = X.
It remains to prove that
1.7. Proposition. The idempotent (F 0 , (Θ)) with F 0 = X is a quasistrong sum of a series of mutually orthogonal projections if and only if there exists a sequence
It remains to prove that
of closed subspaces such that, by putting
It remains to prove that
the series is strongly convergent and f j ∈ F j },
It remains to prove that
. Then the sequence (F i ) satisfies (9)- (11) . Indeed, by the mutual orthogonality of A i , we have F i ⊆ K i , which implies (11), and F i ∩ F i = (Θ) by 1.2. Moreover, for a fixed i and x ∈ X, we have
It remains to prove that
Clearly, (10) holds by 1.2. Let (F j ) ∞ j=1 be a sequence of subspaces satisfying (9)-(11). Then, putting A j = (F j , F j ), we obtain a sequence of mutually orthogonal projections such that q.s.-
It remains to prove that
In the case of the kernel K = (Θ), the situation is more complicated. Let us start with the following 1.8. Definition. Let Y be a linear subspace of X (not closed in general). A sequence (F 1 , F 2 , . . .) of closed subspaces is said to be basic for Y relative to a closed subspace F 0 iff
It remains to prove that
the series is strongly convergent and f j ∈ F j } and (13)
It remains to prove that
where (9) and (10) of Proposition 1.7 are satisfied.
It remains to prove that
If (x j ) is a Schauder basis in X, then the one-dimensional subspaces [x j ] form a basic sequence for X. Conversely, if one-dimensional spaces [x j ] form a basic sequence for X, then (x j ) is a Schauder basis in X. Now, we are in a position to formulate the following characterization of limit idempotents.
It remains to prove that
1.9. Theorem. Let (F 0 , K) be a densely defined idempotent in X. Then the following conditions are equivalent:
It remains to prove that
A i for some mutually orthogonal projections A i ; (ii) there exists a sequence (F j ) of subspaces of X, basic for F 0 relative to K, where K is closed and maximal in the sense that, if (F j ) is basic for F 0 relative to some closed subspace
It remains to prove that
. . : the series is strongly convergent and f i ∈ F i }. Moreover, for F 0 = K, condition (13) is satisfied (comp. the proof of Proposition 1.7), so (F j ) is basic for F 0 relative to K. Let K ′ ⊇ K and let
Convergence of orthogonal series of projections

145
Consequently,
145
(
ii)⇒(i). Let us put
). Obviously, A i are mutually orthogonal. We shall show that q.s.-
ii)⇒(i). Let us put
Indeed, by 1.2,
ii)⇒(i). Let us put
. . : the series is strongly convergent, f i ∈ F i }.
It remains to show that
To this end, let us put
It remains to show that
Then we have K⊂K ′ and, as can easily be checked, the sequence (F 1 , F 2 , . . .) is basic for F 0 relative to K ′ . By (ii), this implies K = K ′ , which ends the proof. 2. Almost sure convergence of orthogonal series of projections in L 2 -spaces 2.1. In this section we are interested in the following kind of convergence for operators acting in X = L 2 (Ω, A, µ). Let (A n ) be a sequence of bounded linear operators in X and let A be linear (bounded or not). We say that A n converge to A almost surely (A n → A a.s.) iff A n f → Af µ-almost everywhere for all f ∈ D(A).
It remains to show that
There are several important results concerning this type of convergence. Let us mention here theorems on martingales, on iterates of conditional expectations, individual ergodic theorems or the results on orthogonal series [6] , [9] .
It remains to show that
Let us start with the following theorem, which is an improvement of our previous results [2] , [8] . 
2.2.
Theorem. Let X be a closed subspace of a separable Hilbert space L 2 (Ω, A, µ) (over an arbitrary measure space (Ω, A, µ)). Let (B k ) be a sequence of finite-dimensional operators in X with B k → B strongly. Then there exists an increasing sequence n(k) of indices such that B n(k) → B a.s. P r o o f. By the separability of L 2 (Ω, A, µ), there exists a sequence (A n ) of finite subfields of A such that the conditional expectations P n = E A n converge strongly and almost surely in L 2 (Ω, A, µ) to the identity operator 1 = E A . Obviously, P n are finite-dimensional orthogonal projections in L 2 (Ω, A, µ). Assume first that the operators B k act in the whole space L 2 (Ω, A, µ). Then C n = B n − P n B → 0 strongly and C n are finite-dimensional. One can define, by induction, sequences n(k) ր ∞ and t(k) ր ∞ satisfying t(1) = 1 and
2.2.
Passing to the general case, assume that B k act in a closed subspace X of L 2 (Ω, A, µ). Let (ϕ s ) be an orthonormal basis in X. Then Q n = n s=1 ϕ s form a sequence of finite-dimensional projections in L 2 (Ω, A, µ). Passing, if necessary, to a subsequence, we can assume by the first part of the proof that Q n → 1 X strongly and almost surely. Now it is enough to repeat the previous argument for P k = Q k .
2.3.
Corollary. Let (A k ) be a sequence of finite-dimensional mutually orthogonal projections (not necessarily selfadjoint) in the separable Hilbert space L 2 (Ω, A, µ). Denote by (S, D(S)) a quasi-strong sum of the series ∞ k=1 A k . Assume that S is a closed operator (but not necessarily densely defined). Then there exists n(k) ր ∞ such that n(k) s=1 A s → S a.s. P r o o f. The corollary is an easy consequence of the previous theorem. Indeed, let
2.3.
the series is strongly convergent and f j ∈ F j }.
2.3.
Since S is closed, F 0 is a closed subspace of L 2 (Ω, A, µ). Putting S n = n j=1 A j , we find that S n f 0 → f 0 strongly for each f 0 ∈ F 0 , and S n are finite-dimensional. By Theorem 2.2, there exists n(k) ր ∞ such that
2.3.
R e m a r k. Theorem 2.2 and Corollary 2.3 can (and should) be treated as generalizations of the classical Marcinkiewicz result [10] .
2.4.
In contrast to the case of quasi-strong convergence, roughly speaking, each unbounded operator A in L 2 (µ) is an almost sure sum of a series of mutually orthogonal projections. The construction of a suitable sequence of projections is based on the following facts:
2.4.
(1) the generalization of Marcinkiewicz's theorem easily gives the existence of a sequence (A j ) of finite-dimensional operators such that Passing to precise formulations, we have the following result.
2.5.
Theorem. Let H = L 2 (Ω, A, µ) be a separable Hilbert space such that 0 < µ(Z n ) → 0 for some (Z n ) ⊂ A. Let A be an unbounded closed and densely defined operator in H. Then there exists a sequence (B j ) of mutually orthogonal finite-dimensional projections in X such that the sums S n = n j=1 B j converge almost surely to A. The following lemma is a key point in the proof of the theorem just formulated. In the sequel, for Z ∈ A, the symbol 1 Z denotes the multiplication operator in L 2 by the indicator χ Z of Z.
2.5.
< ∞} for some spectral measure e(·) satisfying e(Λ, ∞) = 0 for all Λ > 0. Then, for any sequence (A n ) of finite-dimensional operators in H, there exists a sequence (B n ) of mutually orthogonal finite-dimensional projections in H (not necessarily selfadjoint) satisfying, for some
2.5.
can be fixed in such a way that the subspaces
2.5.
are infinite-dimensional. Indeed, one can take, for example, an orthonormal system (ξ s , ζ t ; s, t = 1, 2, . . .) with ξ s ∈ 1 Z(s) H, ζ t ∈ e(t, ∞)H and put
2.5.
Now, fix an orthonormal system (ϕ ij s , ψ ij (Λ); i, j, s, Λ = 1, 2, . . .) satisfying
2.5.
Let us begin the inductive construction of B 1 , B 2 , . . . Put A 0 = B 0 = 0 and assume that the mutually orthogonal projections B 0 , . . . , B n for some n ≥ 0 have already been defined so that
2.5.
. . , n and for some sets Y 1 ⊇ . . . ⊇ Y n in A. Assume additionally that the projections B i for i = 1, . . . , n can be written in the form 
2.5.
We define an operator B n+1 satisfying (16) for k = n + 1 and rewrite the operators B 1 , . . . , B n modifying (17) by taking n + 1 instead of n. Let
2.5.
There exists M ≥ 1, M > M (n) in the case n > 0, such that, for
2.5.
we have
2.5.
Obviously, B 1 , . . . , B n can be written in the form
2.5.
, where
2.5.
Let us fix
2.5.
such that, for
2.5.
we have the estimate
2.5.
Now, one can find a perturbation B n+1 of B n+1 such that B 0 , . . . , B n+1 are the required mutually orthogonal projections. Namely, there exist matrices
2.5.
such that, for an arbitrary orthonormal system (e ij , f
2.5.
the operators 
2.5.
satisfy the conditions C n+1 is a projection, (23)
2.5.
for i = 1, . . . , n. In fact, (23) uniquely determines the entries (a n+1,j ′ j ; j, j ′ = 1, . . . , k(n + 1)), and, for n > 0, condition (24) (respectively (25)) uniquely determines the entries (a i,j
2.5.
Let us remark that, for q > 0, ε > 0, ψ ∈ H, ψ = 1, we see, by (15) , that
2.5.
This makes it possible to find q (large enough) such that
2.5.
(by (14), (18)). Finally, we define B n+1 by putting
2.5.
·, x n+1,j y n+1,j . 
2.5.
To conclude the proof, it remains to show that the operators B 1 , . . . , B n+1 can be rewritten in form (17) with n + 1 instead of n. To this end, it is enough to fix M (n + 1) such that (compare (27)).
2.5.
P r o o f o f T h e o r e m 2.5. For a closed and densely defined A, one can easily construct a sequence (C n ) of finite-dimensional operators such that C n f − Af → 0 for all f ∈ D(A). Let e(·) be the spectral measure of |A|. Putting B = Ì ∞ 0 min(1, λ −1 ) e(dλ), we can apply Theorem 2.2 to a sequence C n B, so there exists n(k) ր ∞ such that C n(k) Bg → ABg for all g ∈ H. Moreover, f ∈ D(A) iff f = Bg for some g. Thus we have S k = C n(k) B → A a.s.
2.5.
Putting A k = S k − S k−1 , S 0 = 0, we have ∞ i=1 A i = A a.s. Now, it is enough to apply Lemma 2.6. R e m a r k. In the theorem just proved, the assumption that H = L 2 (Ω, A, µ) with 0 < µ(Z n ) → 0 for some (Z n ) ⊂ A can be replaced by a weaker condition. Namely, it is enough to assume that H is a closed subspace of L 2 (Ω, A, µ) such that there exists (f n ) ⊂ H with f n = 1, f n → 0 in measure µ.

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