Acta Polytechnica Vol. 51 No. 4/2011

More on PT -Symmetry in (Generalized) Effect Algebras
and Partial Groups

J. Paseka, J. Janda

Abstract

Wecontinue in thedirection of our paper on PT -Symmetry in (Generalized)EffectAlgebras andPartialGroups. Namely
we extend our considerations to the setting ofweakly ordered partial groups. In this setting, any operatorweakly ordered
partial group is a pasting of its partially ordered commutative subgroups of linear operators with a fixed dense domain
over bounded operators. Moreover, applications of our approach for generalized effect algebras are mentioned.

Keywords: (generalized) effect algebra, partially ordered commutative group, weakly ordered partial group, Hilbert
space, (unbounded) linear operators, PT -symmetry, Pseudo-Hermitian Quantum Mechanics.

1 Introduction
It is a well known fact that unbounded linear opera-
tors play the role of the observable in the mathemat-
ical formulation of quantum mechanics. Examples
of such observables corresponding to the momentum
and position observables, respectively, are the follow-
ing self-adjoint unbounded linear operators on the
Hilbert space L2(R):
(i) The differential operator A defined by

(Af )(x) = i
d
dx

f (x)

where i is the imaginary unit and f is a differ-
entiable function with compact support. Then
D(A) �= L2(R), since otherwise the derivative
need not exist.

(ii) (Bf )(x) = xf (x), multiplication by x and again
D(B) �= L2(R), since otherwise xf (x) need not
be square integrable.

Note that in both cases the possible domains are
dense sub-spaces of L2(R), i.e., D(A) = D(B) =
L2(R).

The same is true in general, since for any un-
bounded linear operator there is no standard way to
extend it to the whole space H. By the Hellinger-
Toeplitz theorem, every symmetric operator A with
D(A) = H is bounded.

An important attempt at an alternative formu-
lation of quantum mechanics started in the sem-
inal paper [1] by Bender and Boettcher in 1998.
Bender and others adopted all the axioms of quan-
tum mechanics except the axiom that restricted the
Hamiltonian to be Hermitian. They replaced this
condition with the requirement that the Hamilto-
nian must have an exact PT -symmetry. Later,
A. Mostafazadeh [6] showed that PT -symmetric

quantum mechanics is an example of a more general
class of theories, called Pseudo-Hermitian Quantum
Mechanics.

In [4] Foulis and Bennett introduced the notion
of effect algebras that generalized the algebraic struc-
ture of the set E(H) of Hilbert space effects. In such a
case the set E(H) of effects is the set of all self-adjoint
operators A on a Hilbert space H between the null
operator 0 and the identity operator 1 and endowed
with the partial operation + defined iff A + B is in
E(H), where + is the usual operator sum. Recently,
M. Polakovič and Z. Riečanová [8] established new
examples of generalized effect algebras of positive op-
erators on a Hilbert space.

In [7] we showed how the standard effect alge-
bra E(H) and the latter generalized effect algebras
of positive operators are related to the type of at-
tempt mentioned above. As a by-product, we placed
some of the results from [8] under the common roof
of partially ordered commutative groups.

The aim of the present note is to continue in this
direction.

The paper is organized in the following way. In
Section 1 we recall the basic notions concerning the
theory of (generalized) effect algebras and partially
ordered commutative groups. In Section 2 we show
that the linear operators on H and symmetric linear
operators on H are equipped with the structure of a
weakly ordered commutative partial group. In Sec-
tion 3 we manifest the fact that each of these operator
structures is a pasting of partially ordered commuta-
tive groups of the respective operators with a fixed
dense domain. In the last section we show that our
results concerning the application of renormalization
due to the PT -symmetry of an operator from [7] re-
main true for weakly ordered commutative partial
groups.

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Acta Polytechnica Vol. 51 No. 4/2011

2 Basic definitions and some
known facts

The basic reference for the present text is the clas-
sic book by A. Dvurečenskij and S. Pulmannová [3],
where the interested reader can find unexplained
terms and notation concerning the subject.

We now review some terminology concerning
(generalized) effect algebras and weakly ordered par-
tial commutative groups.
Definition 1 ([4]) A partial algebra (E; +, 0, 1) is
called an effect algebra if 0, 1 are two distinct ele-
ments and + is a partially defined binary operation
on E which satisfy the following conditions for any
x, y, z ∈ E:
(Ei) x + y = y + x if x + y is defined,
(Eii) (x + y) + z = x + (y + z) if one side is defined,
(Eiii) for every x ∈ E there exists a unique y ∈ E

such that x + y = 1 (we put x′ = y),
(Eiv) if 1 + x is defined then x = 0.

Definition 2 ([5]) A partial algebra (E; +, 0) is
called a generalized effect algebra if 0 ∈ E is a distin-
guished element and + is a partially defined binary
operation on E which satisfies the following condi-
tions for any x, y, z ∈ E:
(GEi) x + y = y + x, if one side is defined,
(GEii) (x+y)+z = x+ (y +z), if one side is defined,
(GEiii) x + 0 = x,
(GEiv) x + y = x + z implies y = z (cancellation

law),
(GEv) x + y = 0 implies x = y = 0.

In every generalized effect algebra E the partial
binary operation � and relation ≤ can be defined by
(ED) x ≤ y and y � x = z iff x + z is defined and

x + z = y.
Then ≤ is a partial order on E under which 0 is the
least element of E.

Note that every effect algebra satisfies the axioms
of a generalized effect algebra, and if a generalized
effect algebra has the greatest element then it is an
effect algebra.

Definition 3 A partial algebra (G; +, 0) is called a
commutative partial group if 0 ∈ E is a distinguished
element and + is a partially defined binary operation
on E which satisfy the following conditions for any
x, y, z ∈ E:
(Gi) x + y = y + x if x + y is defined,
(Gii) (x + y) + z = x + (y + z) if both sides are

defined,
(Giii) x + 0 is defined and x + 0 = x,
(Giv) for every x ∈ E there exists a unique y ∈ E

such that x + y = 0 (we put −x = y),
(Gv) x + y = x + z implies y = z.

We will put ⊥G = {(x, y) ∈ G × G | x +
y is defined}.

A commutative partial group (G; +, 0) is called
weakly ordered (shortly a wop-group) with respect
to a reflexive and antisymmetric relation ≤ on G if
≤ is compatible w.r.t. partial addition, i.e., for all
x, y, z ∈ G, x ≤ y and both x + z and y + z are de-
fined implies x+z ≤ y +z. We will denote by P os(G)
the set {x ∈ G | x ≥ 0}.

Recall that wop-groups equipped with a total op-
eration + such that ≤ is an order are exactly partially
ordered commutative groups.

Throughout the paper we assume that H is an
infinite-dimensional complex Hilbert space, i.e., a lin-
ear space with inner product 〈· , ·〉 which is com-
plete in the induced metric. Recall that here for any
x, y ∈ H we have 〈x, y〉 ∈ C (the set of complex num-
bers) such that 〈x, αy + βz〉 = α〈x, y〉+ β〈x, z〉 for all
α, β ∈ C and x, y, z ∈ H. Moreover, 〈x, y〉 = 〈y, x〉
and finally 〈x, x〉 ≥ 0 at which 〈x, x〉 = 0 iff x = 0.
The term dimension of H in the following always
means the Hilbertian dimension defined as the car-
dinality of any orthonormal basis of H (see [2]).

Moreover, we will assume that all considered lin-
ear operators A (i.e., linear maps A : D(A) → H)
have a domain D(A) a linear subspace dense in H
with respect to the metric topology induced by the
inner product, so D(A) = H (we say that A is densely
defined ). We denote by D the set of all dense linear
subspaces of H. Moreover, by positive linear opera-
tors A, (denoted by A ≥ 0) it means that 〈Ax, x〉 ≥ 0
for all x ∈ D(A), therefore operators A are also sym-
metric, i.e., 〈y, Ax〉 = 〈Ay, x〉 for all x, y ∈ D(A) (for
more details see [2]).

To every linear operator A : D(A) → H with
D(A) = H there exists the adjoint operator A∗ of A
such that D(A∗) = {y ∈ H | there exists y∗ ∈ H
such that (y∗, x) = (y, Ax) for every x ∈ D(A)} and
A∗y = y∗ for every y ∈ D(A∗). If A∗ = A then A is
called self-adjoint.

Recall that A : D(A) → H is called a bounded op-
erator if there exists a real constant C ≥ 0 such that
‖Ax‖ ≤ C‖x‖ for all x ∈ D(A) and hence A is an
unbounded operator if to every C ∈ R, C ≥ 0 there
exists xC ∈ D(A) with ‖AxC‖ > C‖xC‖. The set of
all bounded operators on H is denoted by B(H). For
every bounded operator A : D(A) → H densely de-
fined on D(A) = D ⊂ H exists a unique extension B
such as D(B) = H and Ax = Bx for every x ∈ D(A).
We will denote this extension B = Ab (for more de-
tails see [2]). Bounded and symmetric operators are
called Hermitian operators.

We also write, for linear operators A : D(A) → H
and B : D(B) → H, A ⊂ B iff D(A) ⊆ D(B) and
Ax = Bx for every x ∈ D(A).

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Acta Polytechnica Vol. 51 No. 4/2011

3 Operator wop-groups as a
pasting of operator
sub-groups equipped with
the usual sum of operators

Definition 4 Let H be an infinite-dimensional com-
plex Hilbert space. Let us define the following set of
linear operators densely defined in H:

Gr(H) = {A : D(A) → H | D(A) = H

and
D(A) = H if A is bounded}.

Theorem 1 Let H be an infinite-dimensional com-
plex Hilbert space. Let ⊕D be a partial operation on
Gr(H) defined for A, B ∈ Gr(H) by

A ⊕D B =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

A + B (the usual sum)

if A + B is unbounded and

(D(A)= D(B) or

one out of A, B is bounded) ,

(A + B)b if A + B is bounded and

D(A)= D(B),

undefined otherwise

and ≤ be a relation on Gr(H) defined for A, B ∈
Gr(H) by

A ≤ B iff there is a positive linear operator
C ∈ Gr(H) such that B = A ⊕D C.

Then Gr(H) = (Gr(H); ⊕D, 0) is a wop-group with
respect to ≤.

Proof. Let A, B, C ∈ Gr(H). Then (Gi) is valid
since A ⊕D B is defined iff B ⊕D A is defined and
because the usual sum is commutative we get that
A ⊕D B = B ⊕D A. Moreover, (Giii) is valid
since A ⊕D 0 is always defined and A ⊕D 0 = A.
Clearly, (Giv) follows from the fact that for every
A ∈ Gr(H) there exists a unique B ∈ Gr(H) such
that A ⊕D B = 0 (namely we put −A = B and
evidently A + B = 0/D(A) yields 0

b
/D(A) = 0). It re-

mains to check (Gii) and (Gv). This will be proved
by cases. Assume that (A ⊕D B) ⊕D C is defined and
A ⊕D (B ⊕D C) is defined.

First, let (A⊕DB)⊕DC be of the form (A+B)+C,
hence (A + B) + C is unbounded. Then D(A + B) =
D(A)∩D(B) ∈ {D(A), D(B)} and D((A+B)+C) =
D(A + B) ∩ D(C) ∈ {D(A), D(B), D(C)}. Assume
for the moment that D((A + B) + C) = D(A) �= H
(the other cases follow by a symmetric argument).
We have the following possibilities:

(α1): D(A) = D(B), hence A + B is unbounded
and D((A + B) + C) = D(A + B) = D(A) = D(B).
Then either C is unbounded and D(A + B) = D(C)
or C is bounded and D(A + B) ⊂ D(C). In both
cases we have that D(B + C) = D(B) = D(A).
But this yields that (A + B) + C = A + (B + C) =
A ⊕D (B ⊕D C).

(β1): D(A) �= D(B), hence A + B is unbounded,
D(A + B) = D(A) and B is bounded. As in (α1),
either C is unbounded and D(A + B) = D(C) or C is
bounded and D(A+B) ⊂ D(C), D(B) = D(C) = H.
In both cases we have that D(B + C) = D(C) ⊇
D(A). Hence again (A + B) + C = A + (B + C) =
A ⊕D (B ⊕D C).

Similarly, let (A ⊕D B) ⊕D C be of the form
((A + B) + C)b, hence (A + B) + C is bounded.
Because (A + B) is unbounded, C is also unbounded.
Then D(A + B) = D(A) ∩ D(B) ∈ {D(A), D(B)}
and D(A + B) = D(C). Assume that D(A) �= H
(the other case where D(B) �= H is symmetric). We
will distinguish the following cases:

(α2): D(A) = D(B), then D(A + B) = D(A) =
D(B) and because D(C) = D(A + B) we have

D(A) = D(B) = D(C). So ((A + B) + C)b =

(A + (B + C))b = A ⊕D (B ⊕D C).
(β2): D(A) �= D(B), then B is bounded and

D(B) = H, hence D(A) = D(A + B) = D(C).
Then D(B + C) = D(C) = D(A), which yields that

((A + B) + C)b = (A + (B + C))b = A ⊕D (B ⊕D C).
Let (A⊕DB)⊕DC be of the form ((A+B)b+C)b,

hence (A + B)b + C is bounded A + B is bounded
and then C is also bounded. We will verify each of
the following cases:

(α3): D(A) = D(B) �= H, then D(B + C) =
D(B) = D(A). And then ((A + B)b + C)b =

(A + (B + C))b = A ⊕D (B ⊕D C).
(β3): D(A) = D(B) = H, therefore D(B + C) =

H = D(A) and ((A+B)b+C)b = (A+(B+C)b)b =
A ⊕D (B ⊕D C).

And in the last case, let (A ⊕D B) ⊕D C be of the
form ((A + B)b + C). That is, (A + B) is bounded
and C is unbounded. Then we prove:

(α4): D(A) = D(B) = H i.e. A and B are
bounded. Then D((A + B)b + C) = D(C) and
D(C) = D(B + C) = D(A + (B + C)). Hence

((A + B)b + C) = (A + (B + C)) = A ⊕D (B ⊕D C).
(β4): D(A) = D(B) �= H, i.e. A and B are un-

bounded. Then if D(B) �= D(C), ((A⊕D B)⊕D C) =
((A + B)b + C) is defined and D((A + B)b + C) =
D(C), but (B⊕DC) is not defined, so (A⊕D(B⊕DC))
is not defined. In the case that D(B) = D(C) we

have D(B) = D(C) = D(A), so ((A + B)b + C) =
(A + (B + C)) = A ⊕D (B ⊕D C).

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Acta Polytechnica Vol. 51 No. 4/2011

Table 1:

D(C1) D(C2) C1 ⊕D C2 D(C1 ⊕D C2)
= H = H (C1 + C2)b = H

= D(B) = H C1 + C2 = D(C1) = D(B)
= H = D(A) C1 + C2 = D(C2) = D(A)

= D(B) = D(A) = D(C1) C1 + C2 if C1 + C2 is unbounded �= H
(C1 + C2)

b if C1 + C2 is bounded = D(A) = D(B)

Now, assume that A ⊕D B = A ⊕D C. First, as-
sume that A is bounded. Then D(B) = D(C) ⊆
D(A). Hence A + B = A + C. This yields by (Gii)
that C = −A + (A + C) = −A + (A + B) = B. Now,
assume that A is unbounded.

If B is unbounded then D(B) = D(A) and we
will distinguish the following cases:

(γ1): A + B is bounded and hence also A + C is
bounded. It follows that C is unbounded and hence
D(C) = D(A). Therefore also A + B = A + C.

(δ1): A + B is unbounded and hence also A + C
is unbounded. We get that D(C) = D(A), i.e.
A + B = A + C.

In both cases we get as above from (Gii) that
C = B.

If B is bounded we have that A+B is unbounded.
This implies that A + C is unbounded as well. There-
fore D(A + B) = D(A) ⊆ D(B) and D(A) ⊆ D(C).
We then have A + B = A + C. It follows again by
(Gii) that C/D(A) is bounded and B/D(A) = C/D(A).
Therefore B = C.

Let us check that ≤ is reflexive and antisymmet-
ric. Let A, B ∈ Gr(H). Evidently, A ≤ A since
A = A ⊕D 0 and 0 is a positive bounded linear
operator on H. Now, assume that B = A ⊕D C1
and A = B ⊕D C2 for some positive linear operators
C1, C2 on H. By cases we have that (A+C1 = B with
B unbounded or (A + C1)

b = B with B bounded)

and (B+C2 = A with A unbounded or (B+C2)
b = A

with A bounded). Assume first that A+C1 = B with
B unbounded and B + C2 = A with A unbounded.
We have the following possibilities for C1 and C2 (see
Tab. 1).

Now assume that A + C1 = B with B unbounded
and (B + C2)

b = A with A bounded. Then C1 has
to be unbounded with D(C1) = D(B) and C2 can
only also be unbounded with D(C2) = D(B). When

C1 + C2 is bounded then C1⊕D C2 = (C1 + C2)b and
D(C1 ⊕D C2) = H. For C1 + C2 unbounded we have
D(C1 ⊕D C2) = D(B).

The situation for (A + C1)
b = B with B bounded

and (B + C2) = A with A unbounded is symmetric

to the previous case with D(C1 ⊕D C2) = H when
C1 + C2 is bounded and D(C1 ⊕D C2) = D(A) when
C1 + C2 is unbounded.

The last case is (A + C1)
b = B with B bounded

and (B + C2)
b = A with A bounded too. Hence

C1 and C2 are bounded as well and (C1 ⊕D C2) =
(C1 + C2)

b with D(C1 ⊕D C2) = H.
This yields that A ⊕D (C1 ⊕D C2) is defined and

A⊕D (C1⊕D C2) = (A⊕D C1)⊕D C2 = B ⊕D C2 = A.
Hence by (Giii) and (Giv) we obtain that C1⊕D C2 =
0, hence C1 + C2 = 0/D(C1). By [7, Theorem 2] we
have that C1 = C2 = 0.

So, it remains to check that ≤ is compatible with
addition, i.e., for all A, B, C ∈ Gr(H) such that
A ≤ B, C ⊕D A and C ⊕D B are defined we have
that C ⊕D A ≤ C ⊕D B. Again by cases we have
that (C ⊕D A = C + A with C + A unbounded
or C ⊕D A = (C + A)b with C + A bounded)
and (C ⊕D B = C + B with C + B unbounded or
C ⊕D B = (C + B)b with C + B bounded). Since
A ≤ B there is E ∈ Gr(H), E positive such that
A ⊕D E = B. But C ⊕D B = C ⊕D (A ⊕D E).

For the case when E is bounded, clearly, (C ⊕D
A) ⊕D E is always defined. Now assume that E is
unbounded.
In the case when A is unbounded and B is bounded
we get that D(E) = D(A) and D((E + A)b) = H.
We have the following possibilities:
(a): C is bounded. Then D(C + A) = D(A) = D(E).
(b): C is unbounded. Then D(C) = D(A) = D(E).
In the case when both A, B are unbounded we ob-
tain that D(E + A) = D(B) = D(A) = D(E). We
distinguish:
(c): C is bounded. Then D(C + A) = D(A) = D(E).
(d): C is unbounded. Then D(C) = D(A) = D(E).
In the last case assume that A is bounded and B is
unbounded, hence D(E + A) = D(B) = D(E).
(e): C is bounded. Then D(C + A) = H.
(f): C is unbounded. Then D(C + A) = D(C) =
D(B) = D(E).
Hence in all cases (C ⊕D A) ⊕D E is defined.

Recall that we have the following result of [9].

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Acta Polytechnica Vol. 51 No. 4/2011

Theorem 2 [9, Theorem 1] Let H be an infinite-
dimensional complex Hilbert space. Let us define the
following set of positive linear operators densely de-
fined in H:

V(H) = {A : D(A) → H | A ≥ 0,
D(A) = H and
D(A) = H if A is bounded}.

Let ⊕D be defined for A, B ∈ V(H) by A ⊕D B =
A + B (the usual sum) iff

1. either at least one out of A, B is bounded
2. or both A, B are unbounded and D(A) = D(B).

Then VD(H) = (V(H); ⊕D, 0) is a generalized effect
algebra such that ⊕D extends the operation ⊕.

Then P os(Gr(H)) = V(H), hence the generalized
effect algebra V(H) is the positive cone of Gr(H) and,
for all A, B, C ∈ V(H), (A ⊕D B) ⊕D C exists iff
A ⊕D (B ⊕D C) exists.

Definition 5 Let (G, +, 0) be a commutative partial
group and let S be a subset of G such as:
(Si) 0 ∈ S,
(Sii) −x ∈ S for all x ∈ S,
(Siii) for every x, y ∈ S such x + y is defined also

x + y ∈ S.
Then we call S a commutative partial subgroup of G.

Let G be a wop-group with respect to a partial
order ≤G and let ≤S be a partial order on a com-
mutative partial subgroup S ⊆ G. If for all x, y ∈ S
holds: x ≤S y if and only if x ≤G y, we call S a
wop-subgroup of G.

For a commutative partial group G = (G, +, 0)
and a commutative partial subgroup S, we denote
+S = +/S2. We will omit an index and we will write
S = (S, +, 0) instead of (S, +S , 0) where no confusion
can result.

Lemma 1 Let G = (G, +, 0) be a commutative par-
tial group and let S be a commutative partial sub-
group of G. Then (S, +, 0) is a commutative partial
group.

Let G be a wop-group and let S be a wop-subgroup
of G. Then S is a wop-group.

Proof. Conditions (Gi), (Gii) and (Gv) follow im-
mediately from (Siii). Condition (Giii) follows from
(Si) and (Siii) and condition (Giv) follows from (Sii).

Assume now that G is a wop-group such that S
is a wop-subgroup of G. If x, y, z ∈ S, x ≤S y and
x + z, y + z are defined, then x + z, y + z ∈ S and
x + z ≤G y + z hence x + z ≤S y + z.

Lemma 2 Let G = (G, +, 0) be a commutative par-
tial group and S1, S2 commutative partial subgroups
of G. Then S = S1∩S2 is also a commutative partial
subgroup of G.

Proof. Condition (Si) is clear. (Sii): If x ∈ S then
x ∈ S1 and x ∈ S2 hence −x ∈ S1 and −x ∈ S2.
Therefore −x ∈ S.

(Siii): Assume that x, y ∈ S such that x + y is de-
fined. Then x, y ∈ S1 and x, y ∈ S2. Hence x+y ∈ S1
and x + y ∈ S2. This yields x + y ∈ S.

Definition 6 Let G1 = (G1, +1, 01) and G2 =
(G2, +2, 02) be commutative partial groups. A mor-
phism is a map ϕ : G1 → G2 such that, for any
x, y ∈ G1, whenever x +1 y exists then ϕ(x) +2 ϕ(y)
exists, in which case ϕ(x +1 y) = ϕ(x) +2 ϕ(y). If ϕ
is a bijection such that ϕ and ϕ−1 are morphisms we
say that ϕ is an isomorphism of commutative partial
groups and G1 and G2 are isomorphic.

Moreover, let ≤1 on G1 and ≤2 on G2 be par-
tial orders such that G1 and G2 are wop-groups. Let
ϕ : G1 → G2 be a morphism between commutative
partial groups. If for every x, y ∈ G1 : x ≤1 y implies
ϕ(x) ≤2 ϕ(y), then ϕ is a morphism between wop-
groups. If ϕ is a bijection, ϕ and ϕ−1 are morphisms
we say that ϕ is an isomorphism of wop-groups and
G1 and G2 are isomorphic as wop-groups.

Definition 7 Let H be an infinite-dimensional com-
plex Hilbert space. Let us define the following sets
of linear operators densely defined in H:

SGr(H) = {A ∈ Gr(H) | A ⊂ A∗}
HGr(H) = {A ∈ Gr(H) | A ⊂ A∗, D(A) = H}.

i.e. SGr(H) is the set of all symmetric operators and
HGr(H) is the set of all Hermitian operators.

From the definition we can see that HGr(H) ⊆
SGr(H). It is a well known fact that every positive
operator is symmetric and every positive bounded
operator is both self-adjoint and Hermitian (see [2]).

Theorem 3 Let H be an infinite-dimensional com-
plex Hilbert space. Let ≤S be a relation on SGr(H)
defined for A, B ∈ SGr(H) by A ≤S B if and only if
there exists a positive operator C ∈ SGr(H) such as
A ⊕D C = B. Then (SGr(H); ⊕D, 0) equipped with
≤S forms a wop-subgroup of Gr(H).

Proof. Conditions (Si) and (Sii) are clearly satis-
fied. We have to verify that SGr(H) is closed under
addition. Let A, B ∈ SGr(H) be bounded, then they
are Hermitian and it is well known that the sum of
two Hermitian operators is also Hermitian.

Recall that A ⊂ A∗ iff for all x, y ∈ D(A) :
〈x, Ay〉 = 〈Ax, y〉. If A is bounded and B is un-
bounded then D(A + B) = D(B) and, for all x, y ∈
D(B), it holds 〈x, (A + B)y〉 = 〈x, Ay〉 + 〈x, By〉 =

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〈Ax, y〉 + 〈Bx, y〉 = 〈(A + B)x, y〉 hence (A + B) ⊂
(A + B)∗.

A similar argument holds for A unbounded, B un-
bounded and A + B unbounded, where D(A + B) =
D(A) = D(B).

If A and B are unbounded and A + B is bounded,
then, for all x, y ∈ D(A) = D(B), we have 〈x, (A +
B)y〉 = 〈x, Ay〉 + 〈x, By〉 = 〈Ax, y〉 + 〈Bx, y〉 =
〈(A + B)x, y〉. Therefore (A + B) ⊂ (A + B)∗. From
(A + B) ⊂ (A + B)b we get ((A + B)b)∗ ⊂ (A + B)∗

and then H = D((A + B)b∗) = D((A + B)∗). Hence
(A + B)∗ = ((A + B)b)∗. Because (A + B)∗ is sym-

metric one obtains that (A + B)∗ = ((A + B)b)∗ =

(A + B)b. Hence (A + B)b = (A ⊕D B) ∈ SGr(H).
Now let A ⊕D C = B where A, B ∈ SGr(H) and

C ∈ Gr(H), C positive. Since C is a positive operator
we get that C ∈ SGr(H). Therefore ≤S =≤/SGr(H)2.

4 Operator weakly ordered
partial groups as a pasting
of operator sub-groups
equipped with the usual
sum of operators

Let us recall the following theorem from [7] that was
our basic motivation for investigating the set of linear
operators on a Hilbert space.
Theorem 4 Let H be an infinite-dimensional com-
plex Hilbert space and let D ∈ D. Let

LinD(H) = {A : D → H | A
is a linear operator defined on D}.

Then (LinD(H); +, ≤, 0) is a partially ordered com-
mutative group where 0 is the null operator, + is the
usual sum of operators defined on D and ≤ is de-
fined for all A, B ∈ LinD(H) by A ≤ B iff B − A is
positive.

Definition 8 Let H be an infinite-dimensional com-
plex Hilbert space and let D ∈ D. Let

GrD (H) = {A ∈ Gr(H) | D(A) = D or A
is bounded}.

SGrD (H) = {A ∈ SGr(H) | D(A) = D or A
is bounded}.

Now, we are going to show that the set GrD (H)
equipped with the prescription ⊕D = ⊕D/(GrD(H))2
and the relation ≤D=≤/(GrD(H))2 is a partially or-
dered commutative group isomorphic to LinD(H).

Theorem 5 Let H be an infinite-dimensional com-
plex Hilbert space and let D ∈ D. Then GrD (H) =
(GrD (H); ⊕D, 0) with respect to ≤D is a wop-
subgroup of Gr(H) such that the induced operation
⊕D is total. Moreover, GrD (H) is isomorphic to
LinD(H) and hence a partially ordered commutative
group.

Proof. Conditions (Si) and (Sii) are clearly satis-
fied. Let us check condition (Siii).

For A, B ∈ GrD (H), first assume that D(A) =
D(B) ∈ {D, H}. Then D(A) = D(B) = D(A + B) ∈
{D, H}, hence A ⊕D B exists in GrD (H). On the
other hand, let D(A) �= D(B). Then either D(A) ⊂
D(B) = H, in which case D(A) = D(A + B) = D
or D(B) ⊂ D(A) = H with D(B) = D(A + B) = D
hence A ⊕D B also exists in GrD (H). Hence for all
A, B ∈ GrD (H) we have that A ⊕D B ∈ GrD (H) i.e.
⊕D is a total operation on GrD (H).

Now let A ⊕D B = C where A, C ∈ GrD (H) and
B ∈ Gr(H), B positive. Since A ⊕D B is defined and
B is positive we have that B ∈ GrD (H) ∩ V(H).

We can define a map ϕ : LinD → GrD (H) where:

ϕ(A) =

{
A if A is unbounded,

Ab if A is bounded.

For A ∈ LinD unbounded it holds D(A) =
D(ϕ(A)) = D hence ϕ(A) ∈ GrD (H). For A bounded
we have D(ϕ(A)) = D(Ab) = H hence A ∈ GrD (H).

We can define ψ : GrD (H) → LinD as ψ(B) = B
for B unbounded and ψ(B) = B/D for B bounded.
Then clearly ψ ◦ ϕ = idLinD and ϕ ◦ ψ = idGrD(H)
hence ϕ is a bijection.

It is evident that ϕ(0) = 0 and + on LinD is total.
For A, B ∈ LinD, let us assume that:

(a): A, B be bounded. Then ϕ(A + B) = (A +

B)b = Ab + Bb = ϕ(A) ⊕D ϕ(B).
(b): A be bounded, B be unbounded. Then

D(A + B) = D(B) = D and ϕ(A + B) = A + B =

Ab + B = ϕ(A) ⊕D ϕ(B).
(c): A be unbounded, B be unbounded, A + B be

unbounded. Then ϕ(A+B) = A+B = ϕ(A)⊕D ϕ(B)
(d): A be unbounded, B be unbounded, A + B

be bounded. Then ϕ(A + B) = (A + B)b = (ϕ(A) +

ϕ(B))b = ϕ(A) ⊕D ϕ(B).
Now, we should verify order preservation, but it

is clear that ϕ and ψ preserve order.

Theorem 6 Let H be an infinite-dimensional com-
plex Hilbert space and let D ∈ D. Then SGrD (H)
with the induced total operation ⊕D and the induced
partial order ≤SGrD(H) is a wop-subgroup of GrD (H)
and hence a partially ordered commutative subgroup
of GrD (H).

Proof. SGrD (H) is a commutative subgroup
of GrD (H) because of Lemma 2 and SGrD (H) =

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GrD (H) ∩SGr(H). We have to check order preserva-
tion. For any A, B ∈ SGrD(H) such that A ≤GrD(H)
B we have that there exists positive C ∈ GrD (H)
such that A + C = B. Since every positive opera-
tor is symmetric we have that C ∈ SGrD (H). This
yields that ≤SGrD(H)= (≤GrD(H))/SGrD(H)2.

Theorem 7 (The pasting theorem for Gr(H))
Let H be an infinite-dimensional complex Hilbert
space. Then the wop-group Gr(H) pastes their par-
tially ordered commutative subgroups GrD (H), D ⊆
H a dense linear subspace of H, together over B(H),
i.e. GrD1(H) ∩ GrD2(H) = B(H) for every pair
D1, D2 of dense linear subspaces of H, D1 �= D2,
and

Gr(H) =
⋃

{GrD(H) | D ∈ D}.

Proof. Straightforward from definition, for D ∈ D,
every bounded A ∈ Gr(H) lies in GrD (H). For any
unbounded B ∈ Gr(H), B ∈ GrD (H) if and only if
D(B) = D hence there is unique GrD (H) in which
B lies. Hence GrD1 (H) ∩ GrD2 (H) = B(H) for all
D1 �= D2, D1, D2 ∈ D. And because GrD (H) in
which B lies exists for every B ∈ Gr(H), we have
Gr(H) =

⋃
{GrD(H) | D ∈ D}.

Theorem 8 (The pasting theorem for SGr(H))
Let H be an infinite-dimensional complex Hilbert
space. Then the wop-group SGr(H) pastes their
partially ordered commutative subgroups SGrD (H),
D ∈ D, together over HGr(H), i.e., for every pair
D1, D2 of dense linear subspaces of H, D1 �= D2,
SGrD1 (H) ∩ SGrD2(H) = HGr(H) and

SGr(H) =
⋃

{SGrD (H) | D ∈ D}.

Proof. Let D ∈ D. Since SGrD (H) = GrD (H) ∩
SGr(H), with previous theorem

⋃
D∈D

SGrD(H) =

⋃
D∈D

(GrD (H) ∩ SGr(H)) =
( ⋃

D∈D
GrD (H)

)
∩

SGr(H) = Gr(H) ∩ SGr(H) = SGr(H). Similarly we
have SGrD1(H)∩SGrD2(H) = (SGr(H)∩GrD1 (H))∩
(SGr(H) ∩ GrD2(H)) = SGr(H) ∩ (GrD1 (H) ∩
GrD2 (H)) = SGr(H) ∩ B(H) = HGr(H) for all
D1 �= D2, D1, D2 ∈ D.

5 PT -symmetry and related
effect algebras

Let us repeat some of the notions concerning the ba-
sics of PT -symmetry from [7]. Let H be a Hilbert
space equipped with an inner product 〈ψ, φ〉. Let
Ω : H → H be an invertible linear operator. Then

we obtain a new inner product 〈〈 −, − 〉〉 on H which
will have the form:

〈〈ψ, ϕ〉〉 = 〈Ωψ, Ωϕ〉, ∀ψ, ϕ ∈ H.

Clearly, our new inner product space is complete with
respect to 〈〈 −, − 〉〉. Let us denote HΩ the corre-
sponding Hilbert space. Hence Ω : HΩ → H and
Ω−1 : H → HΩ provide a realization of the unitary-
equivalence of the Hilbert spaces HΩ and H.

Let us define a map (·)DΩ : GrD (H) →
GrΩ−1(D)(HΩ) by AΩ = Ω−1 ◦ A ◦ Ω for a linear map
A ∈ GrD (H), D ∈ D. We then have
Proposition 1 [7, Proposition 3] Let H be an
infinite-dimensional complex Hilbert space. Assume
moreover that Ω : H → H is an invertible linear op-
erator and D ∈ D. Then

1. (A)DΩ is a positive operator on HΩ iff A is a
positive operator on H.

2. (·)DΩ is an isomorphism of partially ordered com-
mutative groups.

3. (A)DΩ is a Hermitian operator on HΩ iff A is a
Hermitian operator on H.

4. (I)DΩ = I.

The preceding proposition immediately yields
that

Theorem 9 Let H be an infinite-dimensional com-
plex Hilbert space and Ω : H → H an invertible linear
operator. Then the map (·)Ω : Gr(H) → Gr(HΩ) de-
fined by (A)Ω := (A)

D(A)
Ω for all A ∈ Gr(H) is an

isomorphism of wop-groups.

Proof. It follows from Proposition 1 and Theo-
rem 7.

Corollary 1 Let H be an infinite-dimensional com-
plex Hilbert space and Ω : H → H an invertible lin-
ear operator. Then V(H) and V(HΩ) are isomorphic
generalized effect algebras.

Proof. It follows immediately from the fact that
(·)Ω preserves and reflects positive operators and
from Theorem 9.

We say that an operator H : D → H defined on
a dense linear subspace D of a Hilbert space H is
η+-pseudo-Hermitian and η+ is a metric operator if
η+ : H → H is a positive, Hermitian, invertible, lin-
ear operator such that H∗ = η+Hη

−1
+ (see also [6]).

Theorem 10 Let H be an infinite-dimensional com-
plex Hilbert space, let D ⊆ H be a linear sub-
space dense in H and let H : D → H be a η+-
pseudo-Hermitian operator for some metric operator
η+ : H → H such that η+ = ρ2+. Then

1. Gr(Hρ+ ) and Gr(H) are mutually isomorphic
wop-groups such that H ∈ Gr(Hρ+ ) and H is a
self-adjoint operator with respect to the positive-
definite inner product 〈〈 −, − 〉〉 = 〈ρ+−, ρ+−〉
on Hρ+.

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Acta Polytechnica Vol. 51 No. 4/2011

2. V(H) and V(Hρ+) are mutually isomorphic gen-
eralized effect algebras. If moreover H is a pos-
itive operator with respect to 〈〈 −, − 〉〉 (i.e., its
real spectrum will be contained in the interval
[0, ∞)) then H ∈ V(Hρ+).

Proof. It follows from the above considerations
and [7, Theorem 3].

6 Conclusion
In this paper we have shown that a η+-pseudo-
Hermitian operator for some metric operator η+ of
a quantum system described by a Hilbert space H
yields an isomorphism between the weakly ordered
commutative partial group of linear maps on H and
the weakly ordered commutative partial group of lin-
ear maps on Hρ+. The same applies to the general-
ized effect algebras of positive operators introduced
in [9]. Hence, from the standpoint of (generalized)
effect algebra theory the two representations of our
quantum system coincide.

Acknowledgement

The work of the author was supported by the Min-
istry of Education of the Czech Republic under
project MSM0021622409 and by grant 0964/2009 of
Masaryk University. The second author was sup-
ported by grant 0964/2009 of Masaryk University.

References

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[3] Dvurečenskij, A., Pulmannová, S.: New Trends in
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[4] Foulis, D. J., Bennett, M. K.: Effect algebras and
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[6] Mostafazadeh, A.: Pseudo-Hermitian Represen-
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[7] Paseka, J.: PT-Symmetry in (Generalized) Effect
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[8] Polakovič, M., Riečanová, Z.: Generalized Effect
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[9] Riečanová, Z., Zajac, M., Pulmannová, S.: Ef-
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Jan Paseka
E-mail: paseka@math.muni.cz
Jǐŕı Janda
E-mail: 98599@mail.muni.cz
Department of Mathematics and Statistics
Faculty of Science
Masaryk University
Kotlářská 2, CZ-611 37 Brno

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