Complex Analysis and Operator Theory (2026) 20:78
https://doi.org/10.1007/s11785-026-01935-3
Complex Analysis
and Operator Theory
Inner Products on the Hilbert Space S2 of Hilbert–Schmidt
Operators
Josué I. Rios-Cangas1
Received: 12 February 2025 / Accepted: 1 March 2026 / Published online: 13 March 2026
© The Author(s) 2026
Abstract
This work presents a rigorous characterization of continuous inner products on the
Hilbert space S2 of Hilbert–Schmidt operators. We begin by addressing the general
framework of continuous sesquilinear forms on a Hilbert space H and provide a
characterization of all continuous inner products, by means of positive operators in
B(H). Next, we establish necessary and sufficient conditions for an operator in B(S2 )
to be positive. Identifying a continuous inner product with a positive operator enables
us to rigorously describe inner products on S2 .
Keywords Inner products · Hilbert–Schmidt operators · Non-negative operators
Mathematics Subject Classification Primary 46C50 · Secondary 46C05 · 47A65
1 Introduction
In functional analysis, it is a well-known and useful fact that all norms on a finitedimensional normed linear space are equivalent. However, this is not the case for
infinite-dimensional normed spaces; in fact, one can define uncountably many nonequivalent norms [3]. Notably, some of the simplest and most practical norms are
those induced by inner products. As a matter of fact, all the norms induced by continuous definite inner products on a Hilbert space are complete and equivalent (see
Corollary 2.3). This follows from the fact that all the continuous inner products on a
Hilbert space H are in one-to-one correspondence with the positive linear operators
in B(H) (see Theorem 2.2).
The class of Hilbert–Schmidt operators S2 belongs to the class of compact operators
in B(H) and plays a key role in quantum mechanics, especially in the formulation of
Communicated by Fabrizio Colombo.
B Josué Rios-Cangas
jottsmok@xanum.uam.mx
1
Departamento de Matemáticas, Universidad Autónoma Metropolitana Iztapalapa Campus, San Rafael
Atlixco 186, 09340 Iztapalapa, Mexico City, Mexico
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J. I. Rios-Cangas
non-commutative quantum mechanics [1]. One can endow S2 with a Hilbert space
structure by equipping it with the usual inner product defined by (η, τ ) → tr (η∗ τ ),
where η, τ ∈ S2 . Moreover, it is possible to characterize all continuous sesquilinear
forms ϕ : S2 × S2 → C by
ϕ(η, τ ) = tr η∗ Aτ ,
A ∈ B(S2 ) ,
(1.1)
where ϕ is hermitian if and only if A is selfadjoint. Furthermore, ϕ defines either an
inner product or a definite inner product on S2 if and only if A is positive or positivedefinite, respectively (Lemma 3.2).
The aim of this paper is to present a rigorous characterization of continuous inner
products on S2 , achieved through a fine-tuned analysis of positive operators in B(S2 ).
This approach was rigorously detailed in [4] for the space of complex square matrices,
and this work seeks to extend those results to the infinite-dimensional case. To achieve
this goal, we first exhibit a representation of linear operators and selfadjoint operators
in B(S2 ) (Theorems 3.8 and 3.11). After that, we establish necessary and sufficient
conditions under which an operator in B(S2 ) is positive (Subsections 3.2 and 3.3).
These results allow us to rigorously describe continuous inner products on S2 . Roughly
speaking, any continuous sesquilinear form
ϕ(η, τ ) =
tr η∗ an τ bn , η, τ ∈ S2 ,
n
defines an inner product on S2 , if a1 , . . . ≥ 0 and b1 , . . . > 0 in B(H), with ∩n ker an =
{0} (Theorem 3.21). On the other hand, if ϕ is a continuous definite inner product on
S2 then there exist a1 , . . . ≥ 0, with ∩n ker an = {0}, and positive-definite operators
b1 , . . ., in B(H), such that ϕ satisfies either (Theorem 3.24)
∗
ϕ(η, τ ) = −tr η∗ a1 τ b1 +
tr η an τ bn or
(1.2)
n≥2
ϕ(η, τ ) =
tr η∗ an τ bn .
(1.3)
n
The inner product
if its corresponding operator given in (1.1)
mϕ always satisfies (1.3)
m
m
fulfils Aη =
n=1 ân τ b̂n with {ân }n=1 , {b̂n }n=1 ∈ B(H), for m = 1, 2 (Theorem 3.23). Although the appearance of the negative sign in the first term of (1.2) may
seem counterintuitive, it cannot in general be avoided due to Counterexample 3.15 (cf.
[4, Sec. 4.2]).
For the reader’s convenience, we address the representation of all continuous
sesquilinear forms (including inner products) on a Hilbert space H in Section 2.
This discussion serves as a guideline for Section 3 in which we rigorously characterize continuous sesquilinear forms on S2 . In particular, Subsection 3.2 is devoted
to presenting the necessary conditions while Subsection 3.3 addresses the sufficiency
conditions, both of which determine when an operator in B(S2 ) is non-negative. The
final subsection provides a rigorous characterization of all continuous sesquilinear
forms, including continuous inner products on S2 .
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2 Some Aspects about Continuous Sesquilinear Forms on Hilbert
Spaces
For a separable Hilbert space (H, ·, · ) over C, with inner product anti-linear in its
left argument, a linear operator T : dom T ⊂ H → H is said to be symmetric (or
hermitian) if
f,T f ∈ R,
for all f ∈ dom T .
(2.1)
Besides, T is non-negative (T ≥ 0) if (2.1) is non-negative, and positive (T > 0)
if (2.1) is positive, for all non-zero f ∈ dom T . Moreover, a positive operator T is
called positive-definite if
inf { f , T f
: f ∈ dom T ,
f = 1} > 0 .
Every positive operator is invertible. Besides, the notions of being positive and positivedefinite coincide when the Hilbert space satisfies dim H < +∞.
It is not hard to show that a symmetric operator T with dom T = H is selfadjoint
in B(H) (the class of all bounded operators with domain the whole space H). Besides,
a selfadjoint operator T ∈ B(H) satisfies
m T := inf { f , T f
: f ∈ dom T , f = 1} ∈ R ,
(2.2)
which is called the greatest lower bound (briefly g.l.b.) of T . In this fashion, T ∈ B(H)
is positive-definite if m T > 0.
Recall that a sesquilinear form ϕ : H × H → C is hermitian, non-negative or
positive if ϕ( f , f ) ∈ R, [0, +∞) or (0, +∞), respectively, for all non-zero f ∈ H. It
is positive-definite if inf {ϕ( f , f ) : f ∈ H, f = 1} > 0. Besides, ϕ is continuous
(with respect to ·, · ) if there exists C > 0 for which
|ϕ( f , g)| ≤ C f
g ,
f,g ∈ H.
A sesquilinear form is an inner product if and only if it is positive. In this fashion, we
may refer to a sesquilinear form as a definite inner product whenever it is positivedefinite.
Remark 2.1 Any continuous sesquilinear form ϕ admits the following representations
(cf. [2, Th. 6, Sec. 2.4])
ϕ( f , g) = S f , g ; ϕ( f , g) = f , T g ,
T , S ∈ B(H) .
(2.3)
The representations (2.3) are unique and ϕ = T = S .
Note that ϕ and T in (2.3) are hermitian, non-negative, positive or positive-definite,
simultaneously.
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J. I. Rios-Cangas
Theorem 2.2 Every continuous inner product ϕ on (H, ·, · ) admits the unique representation
ϕ(·, ·) = ·, T · , 0 < T ∈ B(H) , with
ϕ = T .
(2.4)
Besides, the following statements are equivalent:
(i) ϕ is definite.
(ii) ·, · is continuous with respect to ϕ.
(iii) The induced norms
f = f , f 1/2 and
f
ϕ = ϕ( f , f )
1/2
,
f ∈ H,
are equivalent.
(iv) T −1 ∈ B(H) and both T , T −1 are positive-definite.
In such above cases, ϕ is complete, i.e, (H, ϕ) is a Hilbert space.
Proof The first part is straightforward from Remark 2.1. Now:
(i)⇒(ii): if ϕ is definite then T is positive-definite by Remark 2.1 and (2.2) yields
f 2 ≤ ϕ( f , f )/m T for all f ∈ H, with m T > 0. Thus,
| f , g |2 ≤ f
2
g 2≤
1
ϕ( f , f )ϕ(g, g) ,
m 2T
whence it follows that ·, · is continuous with respect to ϕ.
(ii)⇒(iii): the mutual continuity of ·, · and ϕ implies the existence of α, β > 0
for which α · ϕ ≤ · ≤ β · ϕ .
(iii)⇒(iv): the equivalence of the norms and the Cauchy-Schwartz inequality yield
| f , g | ≤ f g ≤ β 2 f ϕ g ϕ , for f , g ∈ H and some β > 0, i.e., ·, · is
continuous with respect to ϕ. Then the first part of the assertion implies that ·, · =
ϕ(·, S·), with 0 < S ∈ B(H). So, ·, · = ϕ(·, S·) = ·, T S· and ϕ(·, ·) = ·, T · =
ϕ(·, ST ·), viz. T −1 = S. Also, the continuity guarantees the existence of C > 0 such
that for a normalized element f ∈ H,
1 = f , f ≤ Cϕ( f , f )2 = C f , T f 2 ,
which implies that T is positive-definite. So, we may take the g.l.b. m T > 0 of T to
compute
2
mT
,
f , T −1 f = T T −1 f , T −1 f ≥ m T T −1 f ≥
T 2
(2.5)
where we have used T −1 f T ≥ T T −1 f = 1. Therefore, (2.5) means that
T −1 is also positive-definite.
(iv)⇒(i): it is simple from Remark 2.1.
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78
For the last part of the assertion, note that T 1/2 , T −1/2 ∈ B(H) are positive-definite
and ran T 1/2 = H. Thus, if { f n } is a Cauchy sequence in (H, ϕ) then so is {T 1/2 f n } in
2
(H, ·, · ), since ϕ(g, g) = T 1/2 g , with g ∈ H. Thus, T 1/2 f n converges to T 1/2 f
for some f ∈ H and
2
ϕ( f n − f , f n − f ) = T 1/2 ( f n − f ) → 0 ,
whence it follows that ϕ is a complete inner product.
The following is straightforward from Theorem 2.2.(iii), which provides a special
property of norms induced by inner products.
Corollary 2.3 All the norms induced by continuous definite inner products on (H, ·, · )
are complete and equivalent.
In what follows, we shall use the univocal correspondence established in Theorem 2.2 to rigorously characterize continuous inner products on the class of
Hilbert–Schmidt operators.
3 Continuous Sesquilinear Forms and Inner Products on S2
3.1 Preliminaries on S2
For the convenience of the reader we repeat some relevant material from [2, Sect. 11.3].
It is well-known that the class of Hilbert–Schmidt operators S2 ⊂ B(H) consists of
those operators η : H → H for which
η 2 :=
1/2
ηu n 2
< +∞ ,
(3.1)
n
being {u n } an orthonormal basis for H, where the sum in (3.1) is independent of the
choice of orthonormal basis.
The space S2 is naturally endowed with a Hilbert structure. For instance, (S2 , ·, · 2 )
is a separable Hilbert space, where
η, τ 2 = tr η∗ τ ,
η, τ ∈ S2 ,
(3.2)
induces the norm given in (3.1) and therefore (S2 , · 2 ) is a Banach space. It is simple
to check that η and η∗ belong to S2 simultaneously and η 2 = η∗ 2 .
To simplify some computations, we shall use the Dirac formalism on rank-one
operators. Namely, for u, v ∈ H we define |u v| ∈ S2 by the rule
|u v| f = v, f u ,
f ∈ H.
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J. I. Rios-Cangas
In this fashion, regarding the canonical basis {en } for H we may consider the canonical
basis {εnm } for S2 , where
εnm := |en em | , satisfies εnm ε jk = δm j εnk ;
εnm , ε jk 2 = δn j δmk ,
being n, m, j, k ∈ N and δmn the Kronecker delta function.
Remark 3.1 Any operator a ∈ B(H) has the following representation
a=
en , aem εnm .
(3.3)
n,m
Indeed, the identity operator
af =
n,m
n εnn in B(H) and the continuity of a yield
εnn aεmm f =
en , aem |en em | f ,
f ∈ H,
n,m
whence it follows (3.3). This representation is known as the matrix representation of
a (cf. [2, Subsect. 2.5.3]).
For a, b ∈ B(H) and η ∈ S2 it follows that bηa ∈ S2 , since one readily checks that
bηa 2 ≤ a
b
η 2,
(3.4)
viz. S2 is an ideal on B(H).
The above material will be used freely in the sequel. For instance, the expression (3.2) allows characterizing continuous sesquilinear forms on S2 , by means of
operators in B(S2 ).
Lemma 3.2 Every continuous sesquilinear form ϕ : S2 × S2 → C admits the representation
ϕ(η, τ ) = tr η∗ Aτ ,
η, τ ∈ S2 ,
which is uniquely determined by A ∈ B(S2 ) and satisfies A = ϕ . Besides,
(i) ϕ is hermitian if and only if A is selfadjoint, and
(ii) ϕ is (definite) inner product on S2 if and only if A is positive (definite).
When ϕ is a continuous definite inner product, then it is complete whose induced norm
is equivalent to (3.1).
Proof It is straightforward from Remark 2.1, Theorem 2.2 and (3.2).
By virtue of Lemma 3.2 we are now motivated to describe linear operators in B(S2 ).
This analysis provides a rigorous characterization of continuous sesquilinear forms
and inner products on S2 .
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3.2 Non-negative Selfadjoint Operators in B(S2 ) (necessity)
Let us first consider the following linear operator in S2 . For fixed subsets {an }, {bn } ⊂
B(H), the linear operator A : dom A ⊂ S2 → S2 given by
Aη =
an ηbn , η ∈ dom A ,
(3.5)
n
is possibly unbounded or non-densely defined on S2 . However, it satisfies the following
properties.
Theorem 3.3 The linear operator (3.5) is:
(i) Symmetric if a1 , . . . , b1 , . . ., are selfadjoint.
(ii) Non-negative if a1 , . . . , b1 , . . . ≥ 0.
(iii) Positive if a1 , . . . > 0 and b1 , . . . ≥ 0 with ∩n ker bn = {0}, or b1 , . . . > 0 and
a1 , . . . ≥ 0 with ∩n ker an = {0}.
(iv) Positive-definite if a1 , . . . , b1 , . . . , are positive-definite, such that their respective
g.l.b.’s (see (2.2)) m a1 , . . . , m b1 , . . . satisfy
m a = inf {m an } > 0 and m b = inf {m bn } > 0 .
n
n
Proof (i): if a1 , . . . , b1 , . . . are selfadjoint, then it follows that
η, Aη 2 =
tr η∗ an ηbn =
tr (an ηbn )∗ η
n
=
n
an ηbn , η 2 = Aη, η 2 ,
n
wherefrom η, Aη 2 ∈ R, i.e., A symmetric.
(ii): if a1 , . . . , b1 , . . . ≥ 0 then
η, Aη 2 =
1/2 1/2
1/2 1/2
η, an an ηbn bn
n
2
=
1/2 1/2 2
an ηbn ≥ 0 .
n
2
(3.6)
(iii): conditions a1 , . . . > 0 and b1 , . . . ≥ 0 with ∩n ker bn = {0} imply from [2,
Lem. 3.2.1.2] and [5, Prop. 1.6] that
⊥ ⊥
spann ran bn = ∩n ran bn
= (∩n ker bn )⊥ = H .
(3.7)
Thus, if a non-zero η ∈ dom A yields η, Aη 2 = 0 then one has by (3.6) that ηbn = 0,
since an is invertible. So, η ran bn = {0} for all n ∈ N and by (3.7),
{0} = η spann ran bn = η(H) ,
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J. I. Rios-Cangas
i.e., η = 0, a contradiction. Therefore, η, Aη 2 > 0. The case when b1 , . . . > 0 and
a1 , . . . ≥ 0 with ∩n ker an = {0} follows interchange the roles of an and bn above,
1/2 1/2 2 1/2
1/2 2
bearing in mind that an ηbn = bn η∗ an in (3.6) and η, η∗ are non-zero
2
2
simultaneously.
(iv): it follows by (3.1) that
1/2 2 ηu n , a j ηu n ≥ m a j η 22 , η ∈ S2 , j ∈ N .
a j η =
2
(3.8)
n
1/2 2
Analogously, b j η ≥ m b j η 22 . Thus, (3.6) and (3.8) imply
2
1/2 2
1/2 1/2 2
η, Aη 2 ≥ a j ηb j ≥ m a j ηb j 2
2
∗ 2
1/2 ∗ 2
= m a j b j η ≥ m a j m b j η 2 ≥ m a m b η 22 ,
2
whence one gets that A is positive-definite.
Remark 3.4 Theorem 3.3.(iii) cannot be relaxed to
a1 , . . . , b1 , . . . ≥ 0 , with
∩n ker an = ∩n ker bn = {0} .
Indeed, εnn ≥ 0 and ∩n ker εnn = {0} for n ∈ N. However, for h, k ∈ N
εhk ,
n
= εhk , εhk εhk 2 = 0 ,
εnn εhk εnn
h = k .
2
One can determine the property to be densely defined of the operator A given in (3.5)
by means of its operators an ’s and bn ’s. Namely, from (3.3)
an =
(n)
αhk εhk and bn =
h,k
r ,m
(n)
(n)
βr(n)
m εr m , with αhk , βr m ∈ C .
In this way we claim that A is a densely defined if and only if
2
(n) (n) α β < +∞ ,
n h j ls h,s
for all j, l ∈ N .
(3.9)
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78
Indeed, it follows for j, l ∈ N that
2
Aε jl 2 =
a
ε
b
e
n
jl
n
s
2
s
n
2
2
(n) (n) (n) (n) =
αh j βls eh α β .
=
n h j ls s
n,h
h,s
Therefore, (3.9) holds true if and only if span{εnm } ⊂ dom A. In this case, S2 =
span{εnm } ⊂ dom A ⊂ S2 , i.e., dom A = S2 , as required.
When A is symmetric and densely defined then its adjoint is well-defined in S2 and
A ⊂ A∗ [2, Sect. 4.1].
Remark 3.5 The operator A satisfying Theorem 3.3.(i)-(iv) is as well selfadjoint in
B(S2 ) if dom A = S2 .
3.3 Non-negative Selfadjoint Operators in B(S2 ) (sufficiency)
We shall first characterize the operators in B(S2 ). Recall that the operators A and A∗
belong to B(S2 ) simultaneously.
Lemma 3.6 Every linear operator A ∈ B(S2 ) is decomposed into
Aη =
εnm ηanm ,
η ∈ S2 ,
(3.10)
n,m
where anm :=
j,k εnk , Aεm j 2 ε jk ∈ B(H).
Proof We first
show that anm ∈ B(H). Since A ∈ B(S2 ), it follows that A∗ ∈ B(S2 )
∗
and A εnk = j,s εs j , A∗ εnk 2 εs j ∈ S2 , wherefrom
(A∗ εnk )∗ =
A∗ εnk , εs j 2 ε js ∈ S2 .
(3.11)
j,s
Thus, for f ∈ H one has f = t f t et with f t ∈ C, and f = k f k εnk 2 ,
∗
which
implies
k f k εnk ∈ S2 . Thereby, A ( k f k εnk ) ∈ S2 ⊂ B(H) and
(A∗ ( k f k εnk ))∗ ∈ B(H), whose continuity and (3.11) yield
anm f =
f t εnk , Aεm j 2 ε jk et =
j,k,t
=
j,s,k
f k εnk , Aεm j 2 e j
j,k
∗
f k A εnk , εs j 2 ε js em =
A
∗
k
∗
f k εnk
em ∈ H ,
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J. I. Rios-Cangas
whence dom anm = H. Now, Aεms =
j,k ε jk , Aεms 2 ε jk and
εr n Aεms = εr s anm , r , s, n, m ∈ N .
(3.12)
Since εr n Aεms , εr s ∈ S2 ⊂ B(H), for a sequence { f t } ⊂ H such that f t → f and
anm f t → g, it follows by (3.12) that
εrr anm f = εr n Aεmr f = lim εr n Aεmr f t
t
= lim εrr anm f t = εrr g.
t
Then, anm f = r εrr anm f = r εrr g = g. Hence, anm is a closed operator defined
in the whole space
H, viz. anm ∈ B(H).
Now, if η = l,s ηls εls ∈ S2 with ηls ∈ C, then
Aη =
l,s,n,k
=
ηls εnk , Aεls 2 εnk =
εnm ηls εls anm =
εnm ηls εls εnk , Aεm j 2 ε jk
n,m,l,s, j,k
εnm ηanm ,
n,m
n,m,l,s
which implies (3.10).
Remark 3.7 The decomposition (3.10) is not unique. Namely, one can also decompose
any linear operator A ∈ B(S2 ) into
Aη =
ânm ηεnm ,
η ∈ S2 ,
(3.13)
n,m
where ânm =
j,k ε jm , Aεkn 2 ε jk ∈ B(H).
The proof of the following assertion is straightforward from Lemma 3.6.
Theorem 3.8 If A ∈ B(S2 ) then there exist {an }, {bn } ⊂ B(H) such that
Aη =
an ηbn ,
η ∈ S2 .
(3.14)
n
Remark 3.9 If (3.14) is such that
Aη =
m
an ηbn ,
(3.15)
n=1
m
m
then {an }m
n=1 and {bn }n=1 can be assumed linearly independent. Indeed, if {an }n=1 is
not linearly independent then it contains a linearly independent subset {an }rn=1 , for
Inner Products on the Hilbert Space S2
which a j =
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78
r
n=1 α jn an for j = r + 1, . . . , m. Thus,
Aη =
r
⎛
m
an η ⎝bn +
⎞
α jn b j ⎠ .
j=r +1
n=1
Thereby, we may suppose that {an }m
there
n=1 in (3.15) is linearly independent and
again
r
,
such
that
b
=
β
exists a linearly independent subset {bn }rn=1 of {bn }m
j
n=1 jn an
n=1
for j = r + 1, . . . , m, and
Aη =
r
⎛
⎝an +
m
⎞
β jn a j ⎠ ηbn ,
j=r +1
n=1
where one simply check that {an +
m
r
j=r +1 β jn a j }n=1 is linearly independent.
In what follows, we shall describe selfadjoint operators in B(S2 ). To perform this,
we first describe the adjoint of an operator in B(S2 ).
Lemma 3.10 For an operator
A in B(S2 ) given by Aη =
B(H), it follows that n an∗ ηbn∗ ∈ S2 and
A∗ η =
an∗ ηbn∗ ,
n an ηbn with {an }, {bn } ⊂
η ∈ S2 .
(3.16)
n
Proof One has that A∗ ∈ B(S2 ) since A ∈ B(S2 ). Besides, it follows for s ∈ N,
η, ρ ∈ S2 that τs = sn=1 an∗ ηbn∗ ∈ S2 and
lim ρ, τs 2 = lim
s
s
s
= Aρ, η 2 = ρ, A∗ η 2 .
an ρbn , η
n=1
(3.17)
2
So, {τs } converges weakly to A∗ η, viz. {τs } ⊂
a bounded sequence and thereby
S2 is
∗ ηb∗ ∈ S . Therefore, (3.17) yields ρ,
∗ ηb∗ = ρ, A∗ η for all ρ ∈ S ,
a
a
2
2
2
n
n 2
n n
n n
which implies (3.16).
Remark 3.5 ensures that A ∈ B(S2 ) is selfadjoint if it can be represented as Aη =
n an ηbn , where a1 , . . . , b1 , . . . ∈ B(H) are selfadjoint. The following asserts the
converse statement.
Theorem 3.11 If A is a selfadjoint operator in B(S2 ) then there exist selfadjoint operators a1 , . . . , b1 , . . . ∈ B(H), such that
Aη =
n
an ηbn ,
η ∈ S2 .
(3.18)
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J. I. Rios-Cangas
Proof It follows by Lemma 3.6 that Aη = n,m εnm ηanm for η ∈ S2 , with anm =
j,k εnk , Aεm j 2 ε jk ∈ B(H). Besides, since A is selfadjoint
∗
anm
=
Aεm j , εnk 2 ε∗jk =
εm j , Aεnk 2 εk j = amn .
(3.19)
1
1 Aη + A∗ η −
εmn ηanm −
εmn ηanm
2
2i n,m
n,m
1 1
(εnm ηanm + εmn ηamn ) −
(εnm ηamn − εmn ηanm )
=
2 n,m
2i n,m
1 + i
1−i
1−i
1+i
=
εnm +
εmn η
anm +
amn ,
2
2
2
2
n,m
(3.20)
j,k
j,k
In this fashion, Lemma 3.10 and (3.19) imply
Aη =
1−i
1−i
1+i
which clearly 1+i
2 εnm + 2 εmn and 2 anm + 2 amn are selfadjoint in B(H). Hence,
one arrives at (3.18) by rearranging and renaming (3.20).
Remark 3.12 One can follow the same reasoning of Remark 3.9 to show that
when (3.18) satisfies
Aη =
m
an ηbn ,
n=1
m
the sets of selfadjoint operators {an }m
n=1 and {bn }n=1 can be
assumed linearly independent. This, bearing in mind that the linear combination m
n=1 αn an is selfadjoint
if and only if αn ∈ R, for n = 1, . . . , m.
The set of selfadjoint operators {ε̂nm } where
ε̂nm :=
1−i
1+i
εnm +
εmn ∈ S2 ,
2
2
(3.21)
is an orthonormal basis for S2 . Indeed, one computes for h, k, n, m ∈ N that
1
i
i
1
εnm , εhk 2 − εnm , εkh 2 + εmn , εhk 2 + εmn , εkh 2
2
2
2
2
1
i
i
1
= δnh δmk − δnk δmh + δmh δnk + δmk δnh = δnh δmk .
2
2
2
2
ε̂nm , ε̂hk 2 =
Inner Products on the Hilbert Space S2
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78
Besides, this orthonormal basis satisfies for n, m ∈ N that
ε̂nn = εnn ;
ran εnn , ran εmm ⊂ ran ε̂nm ;
∩n ker ε̂nm = ∩m ker ε̂nm = {0} .
(3.22)
In what follows, we address the case when A ∈ B(S2 ) is non-negative. First observe
that A satisfies Theorem 3.11 and the natural question arises whether it is possible to
consider a1 , . . . , b1 , . . . non-negative in (3.18).
Theorem 3.13 Let A be a non-zero operator in B(S2 ) given by Aη = aηb, with
a, b ∈ B(H). If A ≥ 0 then there exist â, b̂ ≥ 0 in B(H) such that
Aη = âηb̂ , η ∈ S2 .
(3.23)
Besides, â, b̂ are positive or positive-definite if A is positive or positive-definite, respectively.
Proof One has that a, b = 0 since A = 0, and
0 ≤ |g f |, A |g f | 2 = g, ag f , b f ,
f,g ∈ H.
(3.24)
Thus, for f 0 ∈ H such that f 0 , b f 0 = α = 0, one obtains by (3.24) that g, αag ≥ 0,
i.e., â = αa ≥ 0 and (3.23), where b̂ = b/α ≥ 0. This, since for g0 ∈ H satisfying
g0 , âg0 = β > 0, one computes by (3.24) that β f , b̂ f ≥ 0. The second part
follows from the equality in (3.24).
We obtain an analogous result of the last theorem for the case where A is positivedefinite and consists of exactly two summands.
Theorem 3.14 Let a1 , a2 , b1 , b2 ∈ B(H) and A ∈ B(S2 ) given by
Aη = a1 ηb1 + a2 ηb2 ,
η ∈ S2 .
(3.25)
If A is positive-definite then there exist positive-definite â1 , â2 , b̂1 , b̂2 ∈ B(H) such
that
Aη = â1 ηb̂1 + â2 ηb̂2 ,
η ∈ S2 .
(3.26)
Proof We have a1 , a2 , b1 , b2 = 0 since otherwise one can use Theorem 3.13. In
this fashion, for normalized elements f , g ∈ H and m A the g.l.b. of A, one readily
computes that |g f | 2 = 1 and
0 < m A ≤ |g f |, A |g f | 2 = g, a1 g f , b1 f + g, a2 g f , b2 f .
(3.27)
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J. I. Rios-Cangas
Now, let g0 ∈ H such that α1 = g0 , a1 g0 = 0 and α2 = g0 , a2 g0 . Thus, one has
by (3.27) that f , (α1 b1 + α2 b2 ) f ≥ m A and
Aη =
a1
α2
η(α1 b1 + α2 b2 ) + a2 − a1 ηb2 .
α1
α1
By renaming, one can assume b2 positive-definite in (3.25) as well as b1 by using the
above reasoning.
Thereby, consider t, t0 ≥ 0 such that m b2 −t0 b1 = 0 and m b2 −tb1 > 0, for t ∈
[0, t0 ), which implies the existence of a normalized sequence { f n } ⊂ H such that
f j , b2 − t0 b1 f j → 0. So, Aη = (a1 + t0 a2 )ηb1 + a2 η(b2 − t0 b1 ) and
m A ≤ lim |g f j , A |g f j 2 = lim g, (a1 + t0 a2 )g
j
j
f j , b1 f j ,
whence a1 + t0 a2 is positive-definite, since 0 < m b1 ≤ f j , b1 f j ≤ b1 . Moreover,
one can choose t = t0 − , with small enough, to guarantee that b2 −tb1 and a1 +ta2
are positive-definite and
Aη = (a1 + ta2 )ηb1 + a2 η(b2 − tb1 )
which the renaming allows us to assume a1 , b1 , b2 in (3.25) positive-definite. Finally,
reproducing the above reasoning with the g.l.b.’s m b1 −t0 b2 = 0 and m b1 −tb2 > 0, one
can guarantee that b1 − tb2 and a2 + ta1 are positive-definite and Aη = a1 η(b1 −
tb2 ) + (a2 + ta1 )ηb2 , which yields (3.26).
Based on the last two theorems, one might expect that the operators a1 , . . . , b1 , . . .
in (3.18) could also be considered positive-definite, when A is positive-definite. Nevertheless, this is not always true due to the following.
Counterexample 3.15 [ [4, Sec. 4.2]] Under the condition dim H = 2, it follows that
S2 = span{εnm }2n,m=1 . Consider A ∈ B(S2 ) given by
Aη = (η11 + (1 − t)η22 )ε11 + tη12 ε12 + tη21 ε21 + (η22 + (1 − t)η11 )ε22 ,
where η = 2n,m=1 ηnm εnm ∈ S2 , with ηnm ∈ C, and t ∈ (0, 1/2). Then, the operator
A is positive-definite since it readily follows that
η, Aη 2 = t |η11 |2 + |η12 |2 + |η21 |2 + |η22 |2 + (1 − t) |η11 + η22 |2 .
However, one can follow the same lines of the proof of [4, Ex. 4.4] to show that there
is not exist m ∈ N and non-negative operators a j , b j ∈ B(H), for j = 1, . . . , m, such
that Aη = m
n=1 an ηbn .
As shown in the previous example, Theorems 3.13 and 3.14 cannot be extended to
four or more summands (cf. [4]). However, it is possible to provide conditions on the
operators a1 , . . . , b1 , . . . in the decomposition (3.18).
Inner Products on the Hilbert Space S2
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78
Proposition 3.16 For a non-negative operator A in B(S2 ) there exists a set of selfadjoint operators {anm } ⊂ B(H) for which {ε̂nm } in (3.21) satisfies
Aη =
ε̂nm ηanm ,
η ∈ S2 ,
(3.28)
n,m
where ann ≥ 0 (n ≥ 1). Besides a11 , a22 . . . are positive or positive-definite if A is
positive or positive-definite, respectively, with g.l.b. m ann ≥ m A .
Proof If A is non-negative then so is selfadjoint. Thus, bearing in mind Lemma 3.6
and the decomposition (3.20), one computes that
Aη =
n
εnn ηann +
ε̂nm ηânm ,
(3.29)
n,m
n=m
1+i
since ε̂nn = εnn and ânn = ann , being ânm = 1−i
2 anm + 2 amn ∈ B(H) selfadjoint.
Besides, the canonical basis {en } for H implies e j , ε̂nm e j = 0 whenever n = m.
Thus, for f ∈ H and n, m, j ∈ N,
e j f |, ε̂nm e j f | ânm = e j , ε̂nm e j f , ânm f = 0 , n = m .
(3.30)
2
Thereby, one obtains by (3.29) and (3.30) that
e j f |, A e j f | =
e j , εnn e j f , ann f = f , a j j f ,
2
(3.31)
n
whence it follows that a j j is non-negative. Equation (3.31) also implies that a j j is
positive or positive-definite if A is positive or positive-definite, respectively, with
g.l.b. m a j j ≥ m A . Therefore, (3.29) yields (3.28).
Let us give the following assertion which will be useful in the sequel.
Lemma 3.17 For a positive-definite operator A ∈ B(S2 ) there exist selfadjoint operators b11 , anm ∈ B(H), with n, m ∈ N, such that b11 , a11 and a22 are positive-definite
and
Aη = b11 ηa11 +
ε̂nm ηanm .
(3.32)
n,m
(n,m)=(1,1)
Proof One has from Proposition 3.16 that Aη = n,m ε̂nm ηanm , where bn = ann is
positive-definite with g.l.b. m bn ≥ m A , and anm is selfadjoint, for all n, m ∈ N. Take
t ≥ 0 and t ∈ [0, t0 ) for which m b2 −t0 b1 = 0 and m b2 −tb1 > 0. Thereby, there exist
normalized elements f 1 , . . . ∈ H such that f j , b2 − t0 b1 f j → 0. Thus, one readily
computes that
Aη = (ε11 + t0 ε22 )ηb1 + ε22 η(b2 − t0 b1 ) +
n,m
(n,m)=(1,1),(2,2)
ε̂nm ηanm .
(3.33)
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J. I. Rios-Cangas
In this fashion, for
a normalized g ∈ H one gets by Cauchy–Schwarz inequality that
|g f j , A |g f j 2 ≤ A . Thereby, the dominated convergence theorem and (3.33)
yield
0 < m A ≤ lim |g f j , A |g f j 2
j
= lim g, (ε11 + t0 ε22 )g
j
g, ε̂nm g f j , anm f j
n,m
(n,m)=(1,1),(2,2)
⎛
f j , b1 f j +
⎞
⎟
γnm ε̂nm ⎠ g ,
⎜
= g, ⎝γ11 (ε11 + t0 ε22 ) +
(3.34)
n,m
(n,m)=(1,1),(2,2)
being γn,m = lim j f j , anm f j for n, m ∈ N, with (n, m) = (2, 2), whence γn,m ≤
anm and
γnn ≥ m bn > 0 for n = 2. Thus, one has by (3.34) that γ11 (ε11 +
n,m
γnm ε̂nm is positive-definite in B(H). Besides, we can choose
t0 ε22 )+
(n,m)=(1,1),(2,2)
t= t0 − , with small enough, to guarantee that b2 −tb1 and ε̃11 = γ11 (ε11 +tε22 )+
n,m
γnm ε̂nm are positive-definite. Now we can use this and (3.33) to
(n,m)=(1,1),(2,2)
rewrite A as
Aη = ε̃11 η
b1
+ ε22 η(b2 − tb1 ) +
γ11
n,m
(n,m)=(1,1),(2,2)
γnm
ε̂nm η anm −
b1 ,
γ11
which by renaming one arrives at (3.32).
According to Remark 3.5, if A ∈ B(S2 ) satisfies Aη =
n an ηbn , where
{an }, {bn } ⊂ B(H) meet the conditions of Theorem 3.3.(iv), then A is positive-definite.
Under certain conditions, the following result establishes the converse statement.
Theorem 3.18 If A ∈ B(S2 ) is positive-definite then there exist non-negative operators
a3 , . . . ∈ B(H) and positive-definite operators a1 , a2 , b1 , . . . ∈ B(H) such that
Aη = −a1 ηb1 +
an ηbn ,
η ∈ S2 .
(3.35)
n≥2
Moreover, if there exist ζn > 0 for n = 2, . . . , such that
bn − ζn b1 are positive-definite and − a1 +
ζn a n ≥ 0 ,
(3.36)
n≥2
then there exist non-negative operators â1 , . . . ∈ B(H), with ∩n ker ân = {0}, and
positive-definite operators b̂1 , . . . ∈ B(H) such that
Aη =
n
ân ηb̂n ,
η ∈ S2 .
(3.37)
Inner Products on the Hilbert Space S2
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78
Proof If A is positive-definite then it satisfies Lemm 3.17. So, taking into account the
decomposition (3.32), since b11 , a11 , a22 are positive-definite, one can choose βnm > 0
sufficiently large such that βnm a22 + anm is positive-definite
with n, m ∈ N, (n, m) =
n,m
βnm ε̂nm is
(1, 1), (2, 2). Thus, bearing in mind that b22 = ε22 −
(n,m)=(1,1),(2,2)
selfadjoint,
Aη = b11 ηa11 + b22 ηa22 +
ε̂nm η (βnm a22 + anm )
n,m
(n,m) =(1,1),(2,2)
= b11 η(a11 + αa22 ) − (αb11 − b22 )ηa22 +
ε̂nm η (βnm a22 + anm ) ,
n,m
(n,m) =(1,1),(2,2)
where α > 0 is sufficiently large such that αb11 −b22 is positive-definite. By renaming,
we may suppose that
Aη = −c11 ηa11 + c22 ηa22 +
ε̂nm ηanm ,
(3.38)
n,m
(n,m)=(1,1),(2,2)
where c11 , c22 and anm (n, m ∈ N) are positive-definite and the rest of the operators
selfadjoint. Taking λnm > 0 large enough such that ε̂nm + λnm c11 are non-negative
for n, m ∈ N, n = m, one has by (3.22) and (3.38) that
⎞
⎛
⎟
⎜
λnm anm ⎠ + c22 ηa22
Aη = − c11 η ⎝a11 +
n,m
n=m
+
εnn ηann ,
ε̂nm + λnm c11 ηanm +
n,m
n=m
n≥3
whence by renaming one arrives at (3.35). Condition (3.36) and (3.35) yield
⎛
Aη = ⎝−a1 +
n≥2
⎞
ζn an ⎠ ηb1 +
an η (bn − ζn b1 ) ,
(3.39)
n≥2
where ker (−a1 + n≥2 ζn an ) ∩ ∩n≥2 ker an = {0} since a2 is positive-definite, viz.
invertible. Therefore, (3.39) yields (3.37).
Remark 3.19 It is worth noting that Theorem 3.18 also provides the same decomposition (3.35) with the conditions on {an } and {bn } interchanged. The above follows by
taking the decomposition (3.13) instead of (3.10) in Proposition 3.16 and Lemma 3.17,
which are the basis of Theorem 3.18.
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J. I. Rios-Cangas
3.4 A Rigorous Characterization of Continuous Sesquilinear Forms and Inner
Products on S2
We are now prepared to rigorously characterize the continuous sesquilinear forms on
S2 . In particular, inner products on S2 .
Theorem 3.20 Every continuous sesquilinear form ϕ : S2 × S2 → C admits the representation
ϕ(η, τ ) =
tr η∗ an τ bn ,
{an }, {bn } ⊂ B(H) ,
n
where the linear operator τ → n an τ bn belongs to B(S2 ). Besides, ϕ is hermitian
if and only if a1 . . . , b1 . . . can be chosen as selfadjoint operators.
Proof The first part of the assertion follows from Lemma 3.2 and Theorem 3.8, while
the second part is a consequence of Lemma 3.2.(i) and Theorems 3.3.(i), 3.11.
We now present necessary and sufficient conditions for which a continuous sesquilinear
form defines an inner product on S2 .
Theorem 3.21
Let a1 , . . . > 0 and b1 , . . . ≥ 0 in B(H) with ∩n ker bn = {0}, such
that η → n an ηbn is defined in the whole space S2 . Then
ϕ(η, τ ) =
tr η∗ an τ bn , η, τ ∈ S2 ,
n
defines a continuous inner product on S2 . Besides, ϕ is a complete continuous definite
inner product on S2 if an , bn (n ≥ 1) are positive-definite, with inf n {m an }, inf n {m bn } >
0.
Proof The proof is a direct consequence of Lemma 3.2.(ii), Theorem 3.3.(iii)-(iv) and
Remark 3.5.
Remark 3.22 By virtue of Theorem 3.3.(iii), Theorem 3.21 remains true if the hypothesis of the sets {an } and {bn } are interchanged.
In what follows, we will describe inner products on S2 .
Theorem 3.23 Let ϕ : S2 × S2 → C be an inner product on S2 such that ϕ(η, τ ) =
m
m
m
∗
n=1 tr (η an τ bn ) with {an }n=1 , {bn }n=1 ⊂ B(H), for m = 1, 2.
(i) If m = 1 then there exist positive operators â, b̂ ∈ B(H) such that
ϕ(η, τ ) = tr η∗ âτ b̂ ,
where â, b̂ are as well definite if ϕ is definite.
Inner Products on the Hilbert Space S2
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78
(ii) If m = 2 and ϕ is definite then there exist positive-definite â1 , â2 , b̂1 , b̂2 operators
in B(H) such that
ϕ(η, τ ) = tr η∗ â1 τ b̂1 + tr η∗ â2 τ b̂2 .
Proof Taking into
account (3.4) and the Cauchy-Schwartz inequality, one computes
that |ϕ(η, τ )| ≤ m
bn η 2 τ 2 , i.e., ϕ is continuous. Hence, Lemma 3.2
n=1 an
and Theorems 3.13, 3.14 imply (i) and (ii).
We conclude with the following theorem, which exhibits a general characterization
of continuous definite inner products on S2 .
Theorem 3.24 If ϕ : S2 × S2 → C defines a continuous definite inner product on S2 ,
then there exist non-negative operators a3 , . . . ∈ B(H) and positive-definite operators
a1 , a2 , b1 , . . . ∈ B(H) such that
∗
ϕ(η, τ ) = −tr η∗ a1 τ b1 +
tr η an τ bn ,
η, τ ∈ S2 .
(3.40)
n≥2
Additionally, if there exist ζn > 0, for n = 2, . . . , such that
bn − ζn b1 are positive-definite and − a1 +
ζn an is non-negative ,
n≥2
then there exist non-negative operators â1 , . . . ∈ B(H), with ∩n ker ân = {0}, and
positive-definite operators b̂1 , . . . ∈ B(H) such that
ϕ(η, τ ) =
tr η∗ ân τ b̂n ,
η, τ ∈ S2 .
n
Proof It readily follows from Lemma 3.2.(ii) and Theorem 3.18.
Remark 3.25 Bearing in mind Remark 3.19, Theorem 3.24 gives the same structure (3.40) with the conditions on {an } and {bn } interchanged.
Acknowledgements This work was partially supported by SECIHTI-Mexico: CBF2023-2024-1842, and
UAM-DAI 2024: “Enfoque Analítico-Combinatorio y su Equivalencia de Estados Gaussianos”. The author
is grateful to the anonymous referees for their careful revision of the paper, as their pertinent comments led
to an improved presentation of this work.
Author Contributions The author wrote all the manuscript text and reviewed it.
Funding Open access funding provided by Universidad Autonoma Metropolitana (BIDIUAM).
Data Availability No datasets were generated or analysed during the current study.
Declarations
Competing interests The authors declare no competing interests.
78
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J. I. Rios-Cangas
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