Lobachevskii Journal of Mathematics

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ISSN 1818-9962

Vol. 26, 2007, 91–105 \begin{matrix}c \\ ◯ \end{matrix} Matvejchuk M. S., Ionova A. M.

Matvejchuk M. S., Ionova A. M.

POSITIVE PROJECTIONS AS GENERATORS OF J-PROJECTIONS OF TYPE (B)

(submitted by O. E. Tikhonov)

Abstract. Let A be a von Neumann J-algebra of type (B) acting in an indefinite metric space. The aim of the paper is to study J-projections from A.

In ([1], Chapter XII) the problem of construction of probability theory for quantum mechanics is posed. An analog of boolean algebra of events is quantum logic. An important interpretation of a quantum logic is the set B(H)^{pr} of all orthogonal projections on a Hilbert space H. In construction of measure theory on logics of projections it is important to know the properties and the structure of projections. The problem to construct a quantum field theory sometimes leads to an indefinite metric space ([3]). In indefinite case, the set P of all J-orthogonal projections is an analog of the logics B(H)^{pr}. In the present paper we study Jprojections from von Neumann J-algebras of type (B). The main results of this paper were announced in [6].

space; projection.

2000 Mathematical Subject Classification. 28A60, 81P10, 46C20.

Key words and phrases. Quantum logic; Hilbert space, indefinite metric

1. Introduction

Let H be a complex Hilbert space with an inner product (. , .) and let B(H) be the set of all bounded linear operators in H. Fix a selfadjoint symmetry operator J (J = J^{∗} = J^{−1}, J ≠ ±I). The form [x, y] := (Jx, y) is said to be an indefinite metric, and H with [. , .] is said to be the Krein space (J-space) (see [2]). Put P^{+} := \frac{1}{2}(I + J) and P^{−} := I − P^{+}. Put also Γ^{+} ≡ {f ∈ H : [f, f] = 1} and Γ^{−} ≡ {f ∈ H : [f, f] = −1}. It is clear that JΓ^{±} = Γ^{±}. The set Γ := Γ^{+} ∪ Γ^{−} is an indefinite analog of the unit sphere S of H. Let A ∈ B(H). The operator A^{#} := JA^{∗}J is said to be J-adjoint of A. Note that [Ax, y] = [x, By] for all x, y ∈ H and some B ∈ B(H) if and only if B = A^{#}. An operator A is said to be J-self-adjoint (J-positive, J-negative) if [Ax, y] = [x, Ay] ([Ax, x] ≥ 0, [Ax, x] ≤ 0) for all x, y ∈ H. Note that A is J-self-adjoint (J-positive, J-negative) if and only if JA is self-adjoint (positive, negative, respectively).

An operator p ∈ B(H) is said to be a projection if p^{2} = p. Any one-dimensional projection has the form (., x)y where x, y ∈ H with (y, x) = 1. Let P := {p ∈ B(H) : p^{2} = p = p^{#}}. Thus P is the set of all J-orthogonal (J-self-adjoint) projections in B(H). Any p ∈ P is said to be the J-projection. Let P^{+} (P^{−}) be the set of all J-positive (J-negative, respectively) J-projections. It is clear that P^{+} ∩ P^{−} = {0}. Any onedimensional J-projection has the form p_{f} := [f, f][., f]f, f ∈ Γ. A vector f ∈ Γ^{+} (f ∈ Γ^{−}) if and only if p_{f}, p_{Jf} ∈ P^{+} (∈ P^{−}, respectively).

2. Projections of type (B)

A von Neumann algebra A in H is said to be a von Neumann J-algebra if A ∈ A implies A^{#} ∈ A. Following [4], a commutative von Neumann J-algebra Z is said to be a type (B) algebra if Z contains a pair P, Q ∈ B(H)^{pr} such that P + Q = I, Q^{#} = P. A von Neumann J-algebra A is said to be of type (B) if its center (=A ∩ A′) is of type (B).

Throughout the rest of the paper, A is a von Neumann J-algebra of type (B), a pair P, Q of orthogonal projections in A ∩ A′ satisfying P + Q = I, Q^{#} = P is assumed to be fixed. Set B := P B(H)P + QB(H)Q. It is clear that B is a von Neumann J-algebra of type (B) and A ⊆ B. Put P^{A} = A ∩ P, P^{B} = B ∩ P, and J := P − Q. Clearly,

J Γ^{+} = Γ^{−},  J J = −JJ , i.e., − J = J^{#}.  (1)

Let p, q be projections. Put p ≤ q if pq = qp = p. Note that if p, q ∈ P then pq = p if and only if qp = p. With respect to the standard relations, namely, the ordering ≤, the orthogonal relation p ⊥ q if and only if pq = qp = 0, and the orthocomplementation p ↦ p^{⊥} := I − p, the set P is a quantum logic and P^{A} is a sublogic. Any J-projection from P^{B} is said to be a J-projection of type (B).

Proposition 1. The function σ(P) := J P J is an automorphism of P, for which σ(P^{+}) = P^{−} and σ(P^{+}) = P^{−}.

Proof. Let R ∈ P. Then (σ(R))^{2} = σ(R) and J(σ(R))^{∗}J = J JR^{∗}JJ = σ(R). Thus σ(R) ∈ P. It is clear that σ(σ(R)) = R. Hence σ(P) = P. Furthermore, R_{1} ≤ R_{2} implies σ(R_{1}) ≤ σ(R_{2}), R_{1} ⊥ R_{2} implies σ(R_{1}) ⊥ σ(R_{2}), and σ(R^{⊥}) = (σ(R))^{⊥}.

Now, let R ∈ P^{+}. Then J(J RJ) = −J (JR)J is a negative operator. Hence σ(R) is a J-negative J-projection. By (1), σ(P^{+}) = \frac{1}{2}J (I+J)J = \frac{1}{2}(I + J JJ) = \frac{1}{2}(I − J) = P^{−}. □

We will see below (Theorem 1) that points of P being invariant under σ form the logic P^{B}.

Proposition 2. Let A^{Js} be the set of all J-self-adjoint operators in the von Neumann J-algebra A of type (B) and let A ∈ A. Then A ∈ A^{Js} if and only if A = A_{P} + JA_{P}^{∗} J, where A_{P} := P AP.

Proof. Let A = A^{#}. Then A = AP + AQ = P AP + (JP)(JAJ)(P J) = A_{P} + JA_{P}^{∗} J. Conversely, let A = A_{P} + JA_{P}^{∗} J. Then A^{#} = JA^{∗}J = J(A_{P}^{∗} + JA_{P} J)J = JA_{P}^{∗} J + A_{P} = A. □

Corollary 1. P AP ∩ A^{Js} = {0}, QAQ ∩ A^{Js} = {0}.

Let us denote by L_{P}^{A} the set of all projections from P AP.

Proposition 3. P^{A} = {q + Jq^{∗}J : q ∈ L_{P}^{A}}.

Proof. Let R ∈ P^{A} and q := P R. By Proposition 2, R = P RP + J(P RP)^{∗}J = R_{P} + JR_{P}^{∗} J. Since (R_{P})^{2} + J(R_{P}^{∗})^{2}J = R^{2} = R = R_{P} + JR_{P}^{∗} J, (R_{P})^{2} = R_{P}. Hence R_{P} ∈ L_{P}^{A}.

Conversely, let q ∈ L_{P}^{A}. Then Jq^{∗}J = q^{#} ∈ A. Hence q + Jq^{∗}J ∈ A^{Js}. In addition, (q+Jq^{∗}J)^{2} = q+Jq^{∗}J. By Proposition 2, q+Jq^{∗}J ∈ P^{A}. □

If dim H = ∞ then the logic P^{B} is not a σ-logic (cf. [5, Proposition 2]).

To prove this fact, we will construct a sequence of mutually orthogonal J-projections {R_{n}}_{1}^{∞} ⊂ P^{B} such that the supremum ∑ R_{n} does not exist in P^{B}. Let {ϕ_{n}}_{1}^{∞} be an orthonormal family in P H. (Note that {Jϕ_{n}}_{1}^{∞} is an orthonormal family in QH.) Put f_{k} ≡ (k + 1)^{\frac{1}{2}} ϕ_{2k} + k^{\frac{1}{2}} ϕ_{2k−1} and f_{k}^{−} ≡ (k + 1)^{\frac{1}{2}} ϕ_{2k} − k^{\frac{1}{2}} ϕ_{2k−1}. Then (f_{k}^{−}, f_{k}) = 1, r_{k} := (., f_{k}^{−})f_{k} ∈ L_{P}^{B}. By the construction, {R_{k} := r_{k} + Jr_{k}^{∗}J}_{1}^{∞} is an orthogonal sequence of J-projection (by Proposition 3) from P^{B}.

Assume now that there exists the supremum R := ∑_{1}^{∞} R_{n} ∈ P^{B}. Put

P_{m} ≡ ∑_{1}^{m} R_{k} + ∑_{2m+1}^{∞}((., Jϕ_{n})Jϕ_{n} + (., ϕ_{n})ϕ_{n}).

Then P_{m} ∈ P^{B}, P_{m} ≥ P_{m+1}, ∀m, and P_{m} ≥ R_{n}, ∀m, n. Hence P_{m} ≥

R = ∑_{1}^{∞} R_{n} ≥ ∑_{1}^{m} R_{k}. Thus P_{m} − ∑_{1}^{m} R_{k} ≥ R − ∑_{1}^{m} R_{k} and P_{m} − ∑_{1}^{m} R_{k} ≥

(R − ∑_{1}^{m} R_{k})^{∗}. Finally,

RR^{∗} = (R − ∑_{1}^{m} R_{k})(R − ∑_{1}^{m} R_{k})^{∗} + ∑_{1}^{m} R_{k}R_{k}^{∗}

= (R − ∑_{1}^{m} R_{k})(R − ∑_{1}^{m} R_{k})^{∗} + ∑_{1}^{m}(2k + 1)((., f_{k})f_{k} + (., Jf_{k}^{−})Jf_{k}^{−}), ∀m.

We get a contradiction, since the norm of the right hand side expression tends to infinity when m → ∞.

Theorem 1. Let R be a J-projection. The following conditions are equivalent:

1) R has type (B);

2) R = J RJ;

3) QRP = P RQ = 0.

Proof. Let R have type (B), i. e., R ∈ P^{B}. Then R = P RP + QRQ, hence R = J RJ.

Now, let R = J RJ. Then J R = RJ, hence P R = \frac{1}{2}(I + J)R = \frac{1}{2}(R + RJ) = RP, and consequently, P RQ = RP Q = 0. Similarly, QRP = 0.

Now, let P RQ = QRP = 0. Then R = P RP + QRQ, hence R ∈ P^{B}. □

Corollary 2. Let f ∈ Γ and [f, J f] = 0. Then p_{f} + p_{J f} is a minimal J-projection of type (B).

It appears interesting to compare Theorem 1 with the following proposition.

Proposition 4. 1) If either R ∈ P^{+} or R ∈ P^{−} and either P RQ = 0 or QRP = 0, then R = 0.

2) Let dim H ≥ 4. Then there is R ∈ P such that P RQ = 0, QRP ≠ 0.

3) If R ∈ P and P RQ = 0, then P RP and QRQ are projections.

Proof. 1) Let us consider, for instance, the case R ∈ P^{+} and P RQ = 0. We have 0 = JP RQ = Q(JR)Q. Since JR is a positive operator, the latter implies that P (JR)P = JR. Thus R = JP (JR)P = (JP J)RP = QRP. Finally, R = R^{2} = (QRP)(QRP) = 0.

2) Let dim H ≥ 4. Then we can find φ_{P} ∈ P H ∩ S and φ_{Q} ∈ QH ∩ S such that (φ_{Q}, Jφ_{P}) = 0. Put f_{P} := φ_{P} = g_{P} and f_{Q} := \frac{1}{2}Jφ_{P} + φ_{Q}, g_{Q} := −\frac{1}{2}Jφ_{P} + φ_{Q}. Let us define f = f_{P} + f_{Q}, g = g_{P} + g_{Q}. It is easy to verify that f ∈ Γ^{+}, g ∈ Γ^{−} and [f, g] = 0. Thus p_{f} + p_{g} ∈ P. Finally, P (p_{f} + p_{g})Q = P ([., f]f − [., g]g)Q = ((., Jf_{P})f_{P} − (., Jg_{P})g_{P}) = 0 and Q(p_{f} + p_{g})P = (., Jf_{Q})f_{Q} − (., Jg_{Q})g_{Q} ≠ 0.

3) Now, let P RQ = 0. Then

P RP = (P RP)(P RP) + (P RQ)(QRP) = (P RP)(P RP)

and

QRQ = (P RP)^{#} = (P RP)^{#}(P RP)^{#} = (QRQ)(QRQ).
□

By Theorem 1 and Proposition 4.1), we immediately get

Corollary 3. Let A be a von Neumann J-algebra of type (B). Then P^{A} ∩ P^{+} = {0} = P^{A} ∩ P^{−}.

Proposition 5. Let R ∈ P be an orthoprojection. Then:

1) P^{+}R = RP^{+}, P^{−}R = RP^{−};

2) if, in addition, R has type (B) then J P^{+}RJ = P^{−}R and R = P^{+}R + J (P^{+}R)J.

Proof. 1) Since R ∈ P and R = R^{∗}, we have JR = R^{∗}J = RJ. Hence P^{±}R = RP^{±}.

2) If, in addition, R = J RJ then J P^{+}RJ = (J P^{+}J)J RJ = P^{−}R. This means that R = (P^{+} + P^{−})R = P^{+}R + J (P^{+}R)J. □

Proposition 6. Let q, p ∈ P and let q be a J-projection of type (B). Then p ≤ q if and only if J pJ ≤ q.

Proof. The equality p = qp is equivalent to

J pJ = J qpJ = J pJ (J qJ) = J pJ q.
□

In what follows we will use equations (2) (see below). Let A ∈ A^{Js}. This means that JA = A^{∗}J. Hence

P^{+}AP^{+} = P^{+}(JA)P^{∗} = P^{+}(A^{∗}J)P^{+} = P^{+}A^{∗}P^{+},
P^{−}AP^{+} = −P^{−}(A^{∗}J)P^{+} = −P^{−}A^{∗}P^{+} = −(P^{+}AP^{−})^{∗},  (2)
P^{−}AP^{−} = −P^{−}JAP^{−} = −P^{−}A^{∗}JP^{−} = P^{−}A^{∗}P^{−}.

Consider some properties of projections. Let r be a bounded projection on H. Let us denote by r_{or} the orthogonal projection onto rH ∩ r^{∗}H. By Proposition 4 [5], r_{or} is the greatest orthogonal projection with the property r_{or} ≤ r. A projection r is said to be properly skew projection if r_{or} = 0. Let x, y ∈ H, (x, y) = 1 and ∥x∥ = ∥y∥. Then (., x)y is the properly skew projection if and only if ∥x∥ > 1.

The orthoprojection r_{or} is said to be an orthogonal component of r, and r_{s} := r − r_{or} is said to be a properly skew component of r. It is clear that J r_{or}J (J r_{s}J) is the orthogonal (properly skew, respectively) component of J rJ.

Remark 1. 1) Let r be a bounded projection. Then r ≠ r^{∗} implies that r_{s} is a properly skew projection.

2) R ∈ P^{A} implies R_{or} ∈ P^{A}. Let R = r + Jr^{∗}J, where r ∈ L_{P}^{A}. Then R_{or} = r_{or} + Jr_{or}J.

3) r ∈ L_{P}^{A} is a properly skew projection if and only if R = r + Jr^{∗}J is a properly skew projection.

Now, let r be a properly skew projection. Let us denote by r_{m} the orthogonal projection onto rH. Then r = (r_{m} + r_{m}^{⊥})r(r_{m} + r_{m}^{⊥}) = r_{m} + r_{m}rr_{m}^{⊥}. Let r_{m}rr_{m}^{⊥} = v|r_{m}rr_{m}^{⊥}| be the polar decomposition of r_{m}rr_{m}^{⊥}, where |B| := (B^{∗}B)^{1/2}. Since r is a properly skew projection, vH = r_{m}H. Note that v^{∗}|r_{m}^{⊥}r^{∗}r_{m}| is the polar decomposition of r_{m}^{⊥}r^{∗}r_{m} and |r_{m}^{⊥}r^{∗}r_{m}| = v|r_{m}rr_{m}^{⊥}|v^{∗}. It is clear that

|r − r^{∗}| = |r_{m}rr_{m}^{⊥} − r_{m}^{⊥}r^{∗}r_{m}| = |r_{m}rr_{m}^{⊥}| + |r_{m}^{⊥}r^{∗}r_{m}|,

the cover projection of |r_{m}^{⊥}r^{∗}r_{m}| (of |r_{m}rr_{m}^{⊥}|) is equal to v^{∗}v (vv^{∗}, respectively) and

(v −v^{∗})|r−r^{∗}| = v|r_{m}rr_{m}^{⊥}|−v^{∗}|r_{m}^{⊥}r^{∗}r_{m}| = r_{m}rr_{m}^{⊥} −r_{m}^{⊥}r^{∗}r_{m} = r−r^{∗}. (3)

Put r′_{m} := v^{∗}r_{m}v. By definition, r′_{m}r_{m} = 0 and (v − v^{∗})(r_{m} + r′_{m})(v − v^{∗})^{∗} = r′_{m} + r_{m}. A straightforward verification shows that

(v − v^{∗})[2(r_{m} + r′_{m}) − (r + r^{∗})](v − v^{∗})^{∗} = r + r^{∗}.  (4)

Put x := r + r^{∗} and note that

x^{2} − 2x = (r − r^{∗})^{∗}(r − r^{∗}) = |r − r^{∗}|^{2} ≥ 0.  (5)

Let A_{i} ∈ B(H), i = 1, .., 4, be such that P A_{i}P = A_{i}, for all i. Let us

identify A_{1} + A_{2}J + JA_{3} + JA_{4}J with the matrix (\begin{matrix}A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}). Thus

P^{+} = \frac{1}{2}(I+J) = \frac{1}{2}(I+JP +JQ) = \frac{1}{2}((P +JP J)+JP +P J) = \frac{1}{2} (\begin{matrix}P & P \\ P & P \end{matrix} )

and P^{−} = \frac{1}{2} (\begin{matrix}P & −P \\ −P & P \end{matrix} ). Put X := \frac{1}{4} (\begin{matrix}x & x \\ x & x \end{matrix}).

Theorem 2. Let A be a von Neumann J-algebra of type (B), R = r + Jr^{∗}J, where r ≤ P and r is a properly skew projection from A, and let P^{−}RP^{+} = U|P^{−}RP^{+}| be the polar decomposition for P^{−}RP^{+}. Then X = P^{+}RP^{+}, JUJ = −U and

R = X + U(X^{2} − X)^{1/2} − (X^{2} − X)^{1/2}U^{∗} + U(I − X)U^{∗}. (6)

Conversely, let x ∈ P AP be such that x = x^{∗}, x^{2} − 2x ≥ 0 and w ∈ P AP be a partial isometry with the initial subspace \overline{xH}. Then the

formula (6) defines a J-projection. Here U := \frac{1}{2} (\begin{matrix}w & w \\ −w & −w \end{matrix}). If, in

addition, for x, w the equalities

x = w(2P − x)w^{∗} and w(x^{2} − 2x)^{1/2} = −(x^{2} − 2x)^{1/2}w^{∗} (7)

hold, then R is a J-projection from A.

Proof. A simple matrix computation shows that

P^{+}RP^{+} = X and P^{−}XP^{−} = \frac{1}{4} (\begin{matrix}x & −x \\ −x & x \end{matrix})  (8)

Using (2) with R in place of A and (5) we obtain

|P^{−}RP^{+}| = [(P^{−}RP^{+})^{∗}(P^{−}RP^{+})]^{1/2} = [(−P^{+}RP^{−})(P^{−}RP^{+})]^{1/2} =

[(P^{+}RP^{+})(P^{+}RP^{+})−P^{+}RP^{+}]^{1/2} = (X^{2}−X)^{1/2} = \frac{1}{4} (\begin{matrix}|r − r^{∗}| & |r − r^{∗}| \\ |r − r^{∗}| & |r − r^{∗}| \end{matrix}) .

Put V := \frac{1}{2} (\begin{matrix}(v − v^{∗}) & 0 \\ 0 & −(v − v^{∗}) \end{matrix}) (\begin{matrix}P & P \\ P & P \end{matrix} ) = \frac{1}{2} (\begin{matrix}v − v^{∗} & v − v^{∗} \\ −(v − v^{∗}) & −(v − v^{∗}) \end{matrix}) .

Then

V V^{∗} = \frac{1}{2} (\begin{matrix}(vv^{∗} + v^{∗}v) & −(vv^{∗} + v^{∗}v) \\ −(vv^{∗} + v^{∗}v) & (vv^{∗} + v^{∗}v) \end{matrix})

= \frac{1}{2} (\begin{matrix}(r_{m} + rm^{′}) & −(r_{m} + rm^{′}) \\ −(r_{m} + rm^{′}) & (r_{m} + rm^{′}) \end{matrix}) ≤ P^{−},

and V^{∗}V = \frac{1}{2} (\begin{matrix}(vv^{∗} + v^{∗}v) & (vv^{∗} + v^{∗}v) \\ (vv^{∗} + v^{∗}v) & (vv^{∗} + v^{∗}v) \end{matrix}) ≤ P^{+}. This means that V is a

partial isometry. Since the cover projection of |r −r^{∗}| is equal to r_{m} +rm^{′},

the cover projection of |P^{−}RP^{+}| is equal to \frac{1}{2} (\begin{matrix}(r_{m} + rm^{′}) & (r_{m} + rm^{′}) \\ (r_{m} + rm^{′}) & (r_{m} + rm^{′}) \end{matrix}) .

Hence we have

V (X^{2} − X)^{1/2} = \frac{1}{4} (\begin{matrix}(v − v^{∗}) & 0 \\ 0 & −(v − v^{∗}) \end{matrix}) (\begin{matrix}|r − r^{∗}| & |r − r^{∗}| \\ |r − r^{∗}| & |r − r^{∗}| \end{matrix}) =

by (3) = \frac{1}{4} (\begin{matrix}r − r^{∗} & r − r^{∗} \\ r − r^{∗} & r − r^{∗} \end{matrix}) = P^{−}RP^{+}.

Thus V = U, U(X^{2} − X)^{1/2} = P^{−}RP^{+}, and, by (2),

−(X^{2} − X)^{1/2}U^{∗} = P^{+}RP^{−}.  (9)

Moreover,

U(I − X)U^{∗} = U(P^{+} − X)U^{∗}

= \frac{1}{4} (\begin{matrix}(v − v^{∗}) & 0 \\ 0 & −(v − v^{∗}) \end{matrix}) (\begin{matrix}2P − x & 2P − x \\ 2P − x & 2P − x \end{matrix}) (\begin{matrix}v^{∗} − v & 0 \\ 0 & −(v^{∗} − v) \end{matrix})

= by (4) = \frac{1}{4} (\begin{matrix}x & −x \\ −x & x \end{matrix}) = by (8) = P^{−}RP^{−}. (10)

Summarizing (8), (9), and (10) we have (6). By simple calculations we prove the equality JUJ = −U.

Conversely, let x ∈ P AP be such that x = x^{∗}, x^{2} − 2x ≥ 0 and let w ∈ P AP be a partial isometry with the initial subspace \overline{xH}. Then one can directly show that (6) defines a J-projection.

Now, let (7) hold true for x, w. The matrix entry of R at the first row and the second column is equal to

\frac{1}{4}(x + w(x^{2} − 2x)^{1/2} + (x^{2} − 2x)^{1/2}w^{∗} − w(2P − x)w^{∗}) = 0,

and the matrix entry of R at the first row and the first column is equal to

\frac{1}{4}(x+w(x^{2}−2x)^{1/2}−(x^{2}−2x)^{1/2}w^{∗}+w(2P −x)w^{∗}) = \frac{1}{2}(x+w(x^{2}−2x)^{1/2}).

This means that R = (\begin{matrix}r & 0 \\ 0 & r^{∗} \end{matrix}) is a J-projection from A. Here r :=

\frac{1}{2}(x + w(x^{2} − x)^{1/2}) is a projection from P AP. □

Using Proposition 5.2) we see that (6) is not true if R_{or} ≠ 0.

Note that for any J-projection p there exists (non unique!) representation p = p_{+} + p_{−}, where p_{+} ∈ P^{+}, p_{−} ∈ P^{−}. Let us show that for any R ∈ P^{B} there is a unique special representation of such form.

Let us denote by (pJ)_{+} ((pJ)_{−}) the positive (the negative, respectively) part of self-adjoint operator pJ, p ∈ P.

Lemma 1. If p = p_{+} + p_{−}, p ∈ P, p_{+} ∈ P^{+}, p_{−} ∈ P^{−} and the subspaces p_{+}H, p_{−}H are mutually orthogonal then p_{+} = (pJ)_{+}J and p_{−} = −(pJ)_{−}J.

Proof. By the assumption on p_{+} and p_{−}, we have p_{+}J = (pJ)_{+} and p_{−}J = −(pJ)_{−}. Hence p_{+} = (p_{+}J)J = (pJ)_{+}J and p_{−} = (p_{−}J)J = −(pJ)_{+}J. □

Theorem 3. For any J-projection R of type (B) there exists a unique J-positive J-projection R_{+}^{0} such that R = R_{+}^{0} +J R_{+}^{0}J and the subspaces R_{+}^{0}H, J R_{+}^{0}J H are mutually orthogonal.

Proof. Let us prove that the projection R_{+}^{0} exists.

1) First, let R be an orthogonal J-projection of type (B). The Jprojection P^{+}R (P^{−}R) is J-positive (J-negative, respectively). By Proposition 5.2), R = P^{+}R + J P^{+}RJ. By the construction, the subspaces P^{+}RH and J P^{+}RJ H ⊆ P^{−}H are mutually orthogonal.

2) Now, let R be a properly skew J-projection. By Remark 1.3) and Theorem 2, (6) holds. By Theorem 1 and (10),

J XJ = J P^{+}RP^{+}J = P^{−}(J RJ)P^{−} = U(I − X)U^{∗}.  (11)

Hence J (X^{2} − X)^{1/2}J = U(X^{2} − X)^{1/2}U^{∗}. Thus

(X^{2} − X)^{1/2}J U = J U(X^{2} − X)^{1/2}.  (12)

By Theorem 1 and (9),

J (U(X^{2}−X)^{1/2})J = J (P^{−}RP^{+})J = P^{+}(J RJ)P^{−} = −(X^{2}−X)^{1/2}U^{∗}.

Hence

−(X^{2} − X)^{1/2}U^{∗}J = J U(X^{2} − X)^{1/2}.  (13)

By (12) and (13), J U = −U^{∗}J, i. e.

−U = J U^{∗}J .  (14)

Let us denote by F_{+} (F_{−}) the cover projection of positive (= X_{+}) (negative (= X_{−})) part of self-adjoint operator X, respectively. By (11),

J X_{+}J + J (−X_{−})J = J XJ = U(F_{−} + X_{−})U^{∗} + U(F_{+} − X_{+})U^{∗}. (15)

Since [U(F_{−} + X_{−})U^{∗}][U(F_{+} − X_{+})U^{∗}] = 0,

J X_{+}J = U(F_{−} + X_{−})U^{∗} and − J X_{−}J = U(F_{+} − X_{+})U^{∗}. (16)

By (14) and (16),

J U(X_{+}^{2} − X_{+})^{1/2}J = −U^{∗}J (X_{+}^{2} − X_{+})^{1/2}J =
−U^{∗}U(X_{−}^{2} + X_{−})^{1/2}U^{∗} = −(X_{−}^{2} + X_{−})^{1/2}U^{∗}.  (17)

Put

R_{−}^{0} := −X_{−} + U(X_{−}^{2} + X_{−})^{1/2} − (X_{−}^{2} + X_{−})^{1/2}U^{∗} + U(F_{−} + X_{−})U^{∗},

R_{+}^{0} := X_{+} + U(X_{+}^{2} − X_{+})^{1/2} − (X_{+}^{2} − X_{+})^{1/2}U^{∗} + U(F_{+} − X_{+})U^{∗}. It is easy to verify that R_{±}^{0} ∈ P^{±}, R_{+}^{0}R_{−}^{0} = 0, R = R_{+}^{0} + R_{−}^{0}. Since R_{+}^{0}J is a positive operator, R_{−}^{0}J is a negative operator, and (R_{+}^{0}J)(R_{−}^{0}J) = 0, the subspaces R_{+}^{0}H and R_{−}^{0}H are mutually orthogonal. By (16), (17),

J R_{+}^{0}J = J X_{+}J + J U(X_{+}^{2} − X_{+})^{1/2}J
−J (X_{+}^{2} − X_{+})^{1/2}U^{∗}J + J U(F_{+} − X_{+})U^{∗}J =

U(F_{−} + X_{−})U^{∗} − (X_{−}^{2} + X_{−})^{1/2}U^{∗} + U(X_{−}^{2} + X_{−})^{1/2} − X_{−} = R_{−}^{0} 3) Now, let us consider the general case of R. We have R = R_{or} + R_{s}.

We know that R_{or} is an orthogonal J-projection of type (B), and R_{s} is a properly skew J-projection of type (B). Let (R_{or})_{+} := P^{+}R_{or} and let (R_{s})_{+} be the J-projection from step 2), generated by R_{s}. Put R_{+}^{0} := (R_{or})_{+} + (R_{s})_{+}. Thus the projection R_{+}^{0} is that in question.

By Lemma 1, the J-projection R_{+}^{0} is unique. □

A J-projection R ∈ P^{+} is said to be a generator (for a J-projection R) if R = R + J RJ and the subspaces RH, J RH are mutually orthogonal. Let F^{+} (F^{−}) be the cover projection of P^{+}RP^{+} (of P^{−}RP^{−}, respectively).

Theorem 4. Let R ∈ P^{+} and let P^{−}RP^{+} = U|P^{−}RP^{+}| be the polar decomposition of P^{−}RP^{+}. Then R is a generator if and only if the subspaces J F^{+}H and UH are mutually orthogonal.

Proof. 1) Let R be a generator. Put R := R + J RJ. By Theorem 3, R = R_{+}^{0}. Here R_{+}^{0} is the generator for R from the proof of Theorem 3. From (16) it follows that UH and J F^{+}H are mutually orthogonal.

2) Let us prove some properties. Let R ∈ P^{+}. Then

P^{+}RP^{+} = P^{+}(JR)P^{+} is a positive operator and

P^{−}RP^{−} = −P^{−}(JR)P^{−} is a negative operator.  (18)

Furthermore, P^{+}R^{∗}RP^{+} = P^{+}(JR^{∗})RP^{+} = P^{+}RR^{∗}JP^{+} = P^{+}RR^{∗}P^{+} and P^{−}R^{∗}RP^{−} = P^{−}RR^{∗}P^{−}. We have (P^{+}RP^{+}x, x) ≥ (F^{+}x, x) for all x ∈ H (see Proposition 1, [7]) and

|RP^{+}|^{2} = P^{+}R^{∗}RP^{+} = P^{+}RJRP^{+} =
P^{+}R(2P^{+} − I)RP^{+} = 2(P^{+}RP^{+})^{2} − P^{+}RP^{+}.

a) Hence follows that the initial projection of RP^{+} is equal to the cover projection of P^{+}RP^{+} (i.e., is equal to F^{+}).

In the same way we can prove the following assertion.

b) The final projection of P^{−}R is equal to the cover projection of P^{−}RP^{−} (i.e. F^{−}). We have

(P^{−}RP^{+})(P^{+}R^{∗}P^{−}) = −P^{−}RP^{+}JP^{+}RP^{−} = −P^{−}RP^{+}RP^{−} =

−P^{−}R(I − P^{−})RP^{−} = (P^{−}RP^{−})^{2} − P^{−}RP^{−}.

c) From (18) it follows that the final projection of P^{−}RP^{+} (and hence of U) is equal to the cover projection of P^{−}RP^{−} (i.e., is equal to F^{−})

By b), P^{−}R = F^{−}P^{−}R. By a), RP^{+} = RP^{+}F^{+}. Hence

RP^{+}J RJ = RP^{+}J P^{−}RJ = RP^{+}(F^{+}J F^{−})P^{−}RJ .

Now, let R ∈ P^{+} be a J-projection such that UH and J F^{+}H are mutually orthogonal. Hence 0 = J F^{+}J (UU^{∗}), i.e. by c), F^{+}J F^{−} = 0. Finally, RP^{+}J RJ = RP^{+}(F^{+}J F^{−})P^{−}RJ = 0. Similarly, RP^{−}(J RJ) = 0. Thus R(J RJ) = R(P^{+} + P^{−})J RJ = 0. Hence

R + J RJ ∈ P^{B}, RJ(J RJ) = R(P^{+} − P^{−})(J RJ) = 0.

This means that RH and J RH are mutually orthogonal. Thus R is a generator. □

Let R, R be J-projections of type (B) and R_{+}, R_{+} be their generators. Then RR = 0 (i.e., the subspaces RH, RH are J-mutually orthogonal) implies R + R = (R_{+} + R_{+}) + J (R_{+} + R_{+})J. But the J-positive Jprojection R_{+} + R_{+} is not a generator for R + R, in general.

3. Two-dimensional (minimal) J-projections of type (B)

In what follows we denote by ℜ and ℑ the real and the imaginary part of a complex number.

Let f ∈ H. Put f_{P} := P f, f_{Q} := Qf. In terms of f_{P}, f_{Q} we can give another simple description for vectors from Γ.

Proposition 7. 1) ∥f_{P} ∥ = ∥f_{Q}∥ if and only if (f, J f) = 0 if and only if ℜ(J P^{+}f, P^{−}f) = 0;

2) f ∈ Γ^{+} (∈ Γ^{−}) if and only if 2ℜ[f_{Q}, f_{P}] = 1 (= −1) if and only if 2[f_{Q}, f_{P}] − [f, J f] = 1 (= −1);

3) [f, J f] = 2ℑ[f_{Q}, f_{P}] ∀ f ∈ H.

In connection with Corollary 2 we formulate the following elementary proposition.

Proposition 8. Let f ∈ Γ. The following conditions are equivalent: 1) p_{f}p_{J f} = 0;

2) [f, J f] = 0;

3) ℑ[f_{Q}, f_{P}] = 0;

4) (2[f_{Q}, f_{P}])^{2} = 1;

5) ℑ(J P^{+}f, P^{−}f) = 0.

Lemma 2. Let x, y ∈ P H, (x, y) = 1. The set K := {f ∈ Γ^{+} : [f, J f] = 0, f_{P} = αy, f_{Q} = βJx} is infinite.

Proof. Let α, β be real numbers such that 2βα = 1. Put g_{P} = αy, g_{Q} = βJx and g = g_{P} + g_{Q}. Then 1 = 2βα = 2[g_{Q}, g_{P}] = 2ℜ[g_{Q}, g_{P}]. By Proposition 7 (2), g ∈ Γ^{+}. Since ℑ[g_{Q}, g_{P}] = 0, we have [g, J g] = 0, by Proposition 7 (3). Thus g ∈ K. Since the set \{(α, β) : 2β\overline{α} = 1\} is infinite, the set K is infinite too. □

Thus we have

Corollary 4. For any non-zero J-projection R of type (B) there exists an infinite set of J-positive J-projections R^{0} such that

R = R^{0} + J R^{0}J .

Proof. 1) First, let Q be a two-dimensional J-projection of type (B). Then Q = (., x)y + (., Jy)Jx, where x, y ∈ P H and (x, y) = 1. Let K be from Lemma 2 and let f ∈ K. Since [f, J f] = 0, we have p_{f}p_{J f} = 0, by Proposition 8. This means that p_{f} + p_{J f} is a two-dimensional Jprojection of type (B). Since J f, f ∈ ran(Q), we have Q = p_{f} + p_{J f}. Since K is infinite and p_{f} is a J-positive J-projection, Corollary 4 is true for the case of two-dimensional J-projection.

2) Let us consider the general case. For any non-zero J-projection R of type (B) we can find a two-dimensional J-projection Q of type (B) such that Q ≤ R. Then R−Q is also a J-projection of type (B). Let R_{+}^{0} be the J-positive J-projection such that R_{+}^{0}H, J R_{+}^{0}H are mutually orthogonal (Theorem 3) and R − Q = R_{+}^{0} + J R_{+}^{0}J. The set {R_{+}^{0} + p_{f}: f ∈ K} is infinite. The operator R_{0} := R_{+}^{0} + p_{f}, f ∈ K is a J-positive J-projection and R = R^{0} + J R^{0}J. □

We shall show that there is a unique special f_{0} ∈ K. In order to prove Proposition 11 we need the following proposition.

Proposition 9. Let x, y ∈ P H be such that (x, y) = 1, ∥x∥ = ∥y∥, and let N := {f ∈ Γ^{+} : (f, J f) = [f, J f] = 0 and f_{P} = αy, f_{Q} = βJx}. Then N = \{λf_{0}\}, where f_{0} := \frac{1}{\sqrt{2}}(y + Jx) and |λ| = 1.

Proof. Let f ∈ N. By Proposition 7 (2) and 7 (3),

1 = 2[f_{Q}, f_{P}] = 2β\overline{α}(x, y) = 2β\overline{α}.  (19)

We have 0 = (f, J f) = ∥f_{P} ∥^{2} − ∥f_{Q}∥^{2} = (|α|^{2} − |β|^{2})∥x∥^{2}. Hence |α| = |β|. By (19), α = β and |α| = \frac{1}{\sqrt{2}}. Finally, f = \frac{λ}{\sqrt{2}}(y + Jx), |λ| = 1. □

Note the following. Let x, y ∈ P H be such that (x, y) = 1 and ∥x∥ = ∥y∥ and let f_{0} be from Proposition 9. Then f_{0} ∈ P^{+}H if and only if ∥x∥ = 1 if and only if (., x)y is an orthogonal projection.

A vector f ∈ Γ^{+} is said to be a generator (for a two-dimensional Jprojection R of type (B)) if R = p_{f} + p_{J f} and p_{f}H ⊥ p_{J f}H. It is clear that a vector f ∈ Γ^{+} is a generator if and only if the J-projection p_{f} is a generator.

Proposition 10. The following conditions are equivalent:

1) a vector f is a generator;

2) f ∈ Γ^{+} and (f, J f) = 0 = [f, J f];

3) f ∈ Γ^{+} and (J P^{+}f, P^{−}f) = 0;

4) 2[f_{Q}, f_{P}] = 1 and ∥f_{P} ∥ = ∥f_{Q}∥.

Proof. It is clear that conditions 1) and 2) are equivalent.

2)⇒3). Let (f, J f) = 0 = [f, J f]. By Proposition 7 (1), we have ℜ(J P^{+}f, P^{−}f) = 0. By Proposition 8, ℑ(J P^{+}f, P^{−}f) = 0. Hence (J P^{+}f, P^{−}f) = 0.

3)⇒4). Since f ∈ Γ^{+}, we have 2ℜ[f_{Q}, f_{P}] = 1, by Proposition 7 (2). Since (J P^{+}f, P^{−}f) = 0, we have ℑ[f_{Q}, f_{P}] = 0, by Proposition 8. Hence 2[f_{Q}, f_{P}] = 1. Finally, (J P^{+}f, P^{−}f) = 0 implies (see Proposition 7 (1)) ∥f_{P} ∥ = ∥f_{Q}∥.

4)⇒2). The equality 2[f_{Q}, f_{P}] = 1 implies f ∈ Γ^{+} (see Proposition 7 (2), and by Proposition 8, [f, J f] = 0. By Proposition 7 (1), ∥f_{P} ∥ = ∥f_{Q}∥ implies (f, J f) = 0. □

The following proposition is a two-dimensional analog of Theorem 3 specifying the structure of J-projection R_{+}^{0}.

Proposition 11. Let x, y ∈ P H be such that (x, y) = 1, ∥x∥ = ∥y∥, and let R := (., x)y +(., Jy)Jx be a two-dimensional J-projection of type (B). Then M := {p ∈ P^{+} : R = p + J pJ , (JR)_{+} = Jp, (JR)_{−} = −JJ pJ } = {p_{f_{0}}}, where the vector f_{0} is from Proposition 9.

Proof. Let p ∈ M. By the definition of R, p = p_{f}, where f ∈ Γ^{+}. It is clear that p_{f}(J p_{f}J) = p_{f}p_{J f} = 0. Hence [f, J f] = 0. Since (JR)_{+}(JR)_{−} = 0, 0 = (Jp_{f})(JJ p_{f}J). This means (f, J f) = 0. Since p_{f} < R, f ∈ RH. By the definition of R again, f_{P} = αy, f_{Q} = βJx. By Proposition 9, f = λf_{0}, |λ| = 1. Hence p_{f} = p_{f_{0}}. □

If dim H > 2, then, for any J-projection p_{f}, f ∈ Γ, the set of twodimensional J-projections q such that p_{f} < q is infinite. Therefore the following proposition seems to be interesting.

Proposition 12. For any J-projection p_{f}, f ∈ Γ, there is a unique two-dimensional J-projection R of type (B) such that p_{f} < R.

Proof. Let us consider, for instance, the case f ∈ Γ^{+}. Fix α such that α[f_{Q}, f_{P}] = 1. Put y_{0} := \overline{α}f_{P}, x_{0} := Jf_{Q}. Then (x_{0}, y_{0}) = 1. Thus (., x_{0})y_{0} is a projection. By the construction, (., x_{0})y_{0} ≤ P. Note that

[f, f_{p}] = [f_{P} + f_{Q}, f_{P}] = [f_{Q}, f_{P}], [f, f_{Q}] = [f_{P} , f_{Q}].  (20)

Let us define the two-dimensional J-projection

R := (., x_{0})y_{0} + (., Jy_{0})Jx_{0}

of type (B). From the equality α[f_{Q}, f_{P}] = 1 and (20), it follows that Rp_{f} = p_{f}. Thus p_{f} < R. By Proposition 6, p_{J f} < R. This means that f, J f ∈ RH. Hence the lineal generated by the set {f, J f} is equal to the subspace RH. Any J-projection is uniquely determined by its range space. Therefore R is the unique two-dimensional J-projection with the property p_{f} < R. □

Problem: Does there exist sup{p, J pJ } in P^{B} for any p ∈ P with respect to the order ≤? (cf. Proposition 12).

The author expresses deep gratitude to O.E.Tikhonov for his valuable comments and suggestions.

POSITIVE PROJECTIONS AS GENERATORS 105

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Kazan State University, 420008, Kazan, Universitetskaya 17, Russia E-mail address: Marjan.Matvejchuk@ksu.ru

Ulyanovsk State Pedagogical University, 432000, Ulyanovsk, Tolstogo str. 87, Apartm. 33, Russia

Received June 18, 2007