A Short-type Decomposition Of Forms
aa r X i v : . [ m a t h . F A ] J un A SHORT-TYPE DECOMPOSITION OF FORMS
ZOLT ´AN SEBESTY´EN, ZSIGMOND TARCSAY, AND TAM ´AS TITKOS
Abstract.
The main purpose of this paper is to present a decomposition theoremfor nonnegative sesquilinear forms. The key notion is the short of a form to a linearsubspace. This is a generalization of the well-known operator short defined by M. G.Krein. A decomposition of a form into a shorted part and a singular part (with respectto an other form) will be called short-type decomposition. As applications, we presentsome analogous results for bounded positive operators acting on a Hilbert space; foradditive set functions on a ring of sets; and for representable positive functionals on a ∗ -algebra. Introduction
To begin with we give a brief survey of the required definitions and results from [8],which is our constant reference where the omitted details of this section can be found.Let X be a complex linear space and let t be nonnegative sesquilinear form on it.That is, t is a mapping from the Cartesian product X × X to C , which is linear in thefirst argument, antilinear in the second argument, and the corresponding quadratic form t [ · ] : X → R ∀ x ∈ X : t [ x ] := t ( x, x )is nonnegative. In this paper all sesquilinear forms are assumed to be nonnegative, hencewe write shortly form . The quadratic form of a form fulfills the parallelogram law ∀ x, y ∈ X : t [ x + y ] + t [ x − y ] = 2( t [ x ] + t [ y ]) . According to the Jordan-von Neumann theorem [26, Satz 1.3], a form is uniquely deter-mined via its quadratic form, namely ∀ x, y ∈ X : t ( x, y ) = 14 X k =0 i k t [ x + i k y ] . The set F + ( X ) of forms is partially ordered with respect to the ordering t ≤ w ⇐⇒ ∀ x ∈ X : t [ x ] ≤ w [ x ] . If there exists a constant c such that t ≤ c · w then we say that t is dominated by w ( t ≤ d w , in symbols). Since the square root of the quadratic form defines a seminorm on X , then the kernel of t ker t := (cid:8) x ∈ X (cid:12)(cid:12) t [ x ] = 0 (cid:9) is a linear subspace of X . The Hilbert space H t denotes the completion of the innerproduct space X / ker t equipped with the natural inner product ∀ x, y ∈ X : ( x + ker t | y + ker t ) t := t ( x, y ) . Mathematics Subject Classification.
Primary 47A07, Secondary 47B65, 28A12, 46L51.
Key words and phrases.
Lebesgue decomposition, nonnegative forms, positive operators, absolute con-tinuity, singularity, generalized short.
We say that the form t is strongly w -absolutely continuous ( t ≪ s w ), if ∀ ( x n ) n ∈ N ∈ X N : (cid:0) ( t [ x n − x m ] → ∧ ( w [ x n ] → (cid:1) = ⇒ t [ x n ] → . Remark that this notion is called closability in [8]; cf. also [18]. The singularity of t and w (denoted by t ⊥ w ) means that ∀ s ∈ F + ( X ) : (cid:0) ( s ≤ t ) ∧ ( s ≤ w ) (cid:1) = ⇒ s = 0 . The parallel sum t : w of t and w , and the strongly absolutely continuous (or closable)part D w t of t with respect to w are defined by ∀ x ∈ X : ( t : w )[ x ] := inf y ∈ X (cid:8) t [ x − y ] + w [ y ] (cid:9) and D w t := sup n ∈ N ( t : n w ) . The following decomposition theorem of S. Hassi, Z. Sebesty´en, and H. de Snoo generalizesthe operator decomposition of T. Ando [3, 23], the Lebesgue decomposition of finitelyadditive set functions [4] (see also [16, 22, 25]), and the canonical decomposition of denselydefined forms [18].
Theorem 0.1.
Let t and w be forms on the complex linear space X . Then the decompo-sition t = D w t + ( t − D w t ) is a ( ≪ s , ⊥ ) -type decomposition of the form t with respect to w . That is, D w t is strongly w -absolutely continuous, ( t − D w t ) is w -singular. Furthermore, this decomposition is extremalin the following sense: ∀ s ∈ F + ( X ) : (cid:0) ( s ≤ t ) ∧ ( s ≪ s w ) (cid:1) = ⇒ s ≤ D w t . The decomposition is unique precisely when D w t is dominated by w . For the proof see [8, Theorem 2.11, Theorem 3.8, Theorem 4.6] or [16, Theorem 2.3].It is a natural idea to consider the following notion of absolute continuity: we say that t is w -absolutely continuous ( t ≪ w ) if ker w ⊆ ker t , that is to say, ∀ x ∈ X : w [ x ] = 0 = ⇒ t [ x ] = 0in analogy with the well-known measure case.The setup of this paper is the following. Our main purpose is to present an ( ≪ , ⊥ )-type decomposition theorem for forms which we shall call a short-type decomposition.More precisely, for every pair of forms t and w we shall show that t splits into absolutelycontinuous and singular parts with respect to w , where the absolutely continuous part isextremal in a certain sense. This will be done in Section 1. The key notion is the short ofa form , which is motivated by [2, Theorem 6] of W. N. Anderson and G. E. Trapp.In Section 2 we shall see that this is a generalization of the well-known operator shortdefined by M. G. Krein [11]. Moreover, we present a factor decomposition for the shortedoperator. As an application, we gain also a short-type decomposition on the set of boundedpositive operators (analogous results for matrices can be found in [1, 13]). That is, forevery A, B ∈ B + ( H ) there exist S, T ∈ B + ( H ) such that A = S + T, SHORT-TYPE DECOMPOSITION OF FORMS 3 where ran S ⊆ ran B and ran T / ∩ ran B / = { } . Furthermore, we prove the following characterization: the range of the bounded positiveoperator B is closed if and only if for every A ∈ B + ( H ) the short-type decompositionabove is unique. In this case, the shorted part of A is closable with respect to B .Another important application can be found in Section 3. Using our main result, wewill prove a decomposition theorem for additive set functions. In the σ -additive case thisdecomposition coincides with the well-known Lebesgue decomposition of measures, but inthe finitely additive case it differs from the Lebesgue-Darst decomposition [4]. This factwill demonstrate that the Lebesgue-type decomposition, and the short-type decompositionare different in general, and hence, absolute continuity does not implies strong absolutecontinuity (see also [7, Example 2]).Finally, in Section 4, we apply our result to present a short-type decomposition forrepresentable positive functionals of a ∗ -algebra. We emphasize that we do not make anyassumptions for the algebra, neither the commutativity, nor the existence of unit element.1. A short-type decomposition theorem for forms
Let t and w be forms on the complex linear space X . The purpose of this section isto show that t has a decomposition into a w -absolutely continuous and a w -singularpart. This type decomposition will be called short-type decomposition , or ( ≪ , ⊥ )-typedecomposition. In our further considerations an essential role will be played by the conceptof the short of a form, which is introduced in the following lemma. Lemma 1.1.
Let Y ⊆ X be a linear subspace, and let t ∈ F + ( X ) . Then the followingformula defines a form on X ∀ x ∈ X : t Y [ x ] := inf y ∈ Y t [ x − y ] . Furthermore, t Y is the maximum of the set (cid:8) s ∈ F + ( X ) (cid:12)(cid:12) ( s ≤ t ) ∧ ( Y ⊆ ker s ) (cid:9) . Proof.
Let Y t be the following subspace of H t Y t := (cid:8) y + ker t (cid:12)(cid:12) y ∈ Y (cid:9) and consider the orthogonal projection P from H t onto Y t (the closure of Y t ). Then forall x ∈ X (cid:13)(cid:13) ( I − P )( x + ker t ) (cid:13)(cid:13) t = dist ( x + ker t , Y t ) = inf y ∈ Y (cid:13)(cid:13) ( x − y ) + ker t (cid:13)(cid:13) t = inf y ∈ Y t [ x − y ] . Consequently, t Y is a form, indeed, and Y ⊆ ker t Y . To show the maximality, assume thatthe quadratic form of s vanishes on Y and s ≤ t . According to the triangle inequality wehave s [ x ] ≤ s [ x − y ] ≤ t [ x − y ]for all y ∈ Y , and hence, s [ x ] ≤ inf y ∈ Y t [ x − y ] = t Y [ x ] . (cid:3) Z. SEBESTY´EN, ZS. TARCSAY, AND T. TITKOS
The form t Y is called the short of the form t to the subspace Y .It follows from the definition that if t and w are forms and Y and Z are linear subspaces,then (cid:0) ( t ≤ w ) ∧ Y ⊆ Z (cid:1) = ⇒ t Z ≤ w Y . Now, we are in position to state and prove the main result of this section.
Theorem 1.2.
Let t , w ∈ F + ( X ) be forms. Then there exists a ( ≪ , ⊥ ) -type decompositionof t with respect to w . Namely, t = t ker w + ( t − t ker w ) , where the first summand is w -absolutely continuous and the second one is w -singular.Furthermore, t ker w is the maximum of the set (cid:8) s ∈ F + ( X ) (cid:12)(cid:12) ( s ≤ t ) ∧ ( s ≪ w ) (cid:9) . The decomposition is unique precisely when t ker w is dominated by w .Proof. It follows from the previous lemma that t ker w ≪ w , and that t ker w is maximal. Let s be a form such that s ≤ w and s ≤ t − t ker w . Since t ker w ≤ t ker w + s ≤ t and the quadraticform of t ker w + s vanishes on ker w , the maximality of t ker w implies that s = 0.It remains only to prove that the decomposition is unique if and only if t ker w is dominatedby w . Let c be a constant such that t ≤ c · w (we may assume that c >
1) and let t = t + t be an ( ≪ , ⊥ )-type decomposition. Since t ker w is maximal, we have t = t − t ≥ t ker w − t ≥ c ( t ker w − t ) and w ≥ c t ker w ≥ c ( t ker w − t )which is a contradiction, unless t ker w = t . Finally, observe that D w t ≤ t ker w , and therefore,every ( ≪ s , ⊥ )-type decomposition is a ( ≪ , ⊥ )-type decomposition as well. Indeed, t ker w [ x ] = inf y ∈ ker w t [ x − y ] = inf y ∈ ker w { t [ x − y ] + n w [ y ] } ≥ inf y ∈ X { t [ x − y ] + n w [ y ] } = ( t : n w )[ x ]holds for all n ∈ N and x ∈ X , therefore D w t = sup n ∈ N ( t : n w ) ≤ t ker w . Thus if the ( ≪ , ⊥ )-type decomposition is unique, then t ker w = D w t , and t ker w ≤ d w according to Theorem 0.1. (cid:3) Observe that ( t Y ) Y = t Y for each subspace Y , i.e., shortening to a subspace is anidempotent operation. Furthermore, t ≪ w precisely when t ker w = t . Remark 1.3.
Let A be a complex algebra, let I ⊆ A be a left ideal, and let t be a representable form on A . That is, a nonnegative sesquilinear form, which satisfies ( ∀ a ∈ A ) ( ∃ λ a >
0) ( ∀ b ∈ A ) : t [ ab ] ≤ λ a t [ b ] . A simple observation shows that t I is representable t I [ ab ] = inf x ∈ I t [ ab − x ] ≤ inf x ∈ I t [ ab − ax ] ≤ inf x ∈ I λ a t [ b − x ] = λ a t I [ b ] . If w is a representable form on A as well, then ker w is obviously a left ideal, and hencewe have the following decomposition t = t ker w + ( t − t ker w ) where t ker w ≪ w , ( t − t ker w ) ⊥ w , and t ker w is representable. For a Lebesgue-type decom-position of representable forms we refer the reader to [20] . SHORT-TYPE DECOMPOSITION OF FORMS 5
Finally, we show that the shorted form t Y possesses an extremal property. In fact, weprove that t Y is a disjoint part of t for every subspace Y , or equivalently, t Y is a so-called t -quasi unit. After recalling the corresponding definitions, in Lemma 1.4 we give acharacterization of the extremal points of the convex set[0 , t ] = (cid:8) w ∈ F + ( X ) (cid:12)(cid:12) w ≤ t (cid:9) . We say that u is a t -quasi-unit , if D u t = u . The form u is a disjoint part of t if u and t − u are singular. The set of extremal points of a convex set C is denoted by ex C . Forthe terminology see [6, 12, 17]. Lemma 1.4.
Let t and u be forms on D such that u ≤ t , and let λ > and µ > bearbitrary real numbers. Then the following statements are equivalent. ( i ) u is a t -quasi-unit, i.e., D u t = u . ( ii ) There exists w such that u = D w t . ( iii ) u is a disjoint part of t . ( iv ) u ∈ ex[0 , t ] . ( v ) ( λ u ) : ( µ t ) = λµλ + µ u . ( vi ) ( λ u ) : t = u : ( λ t ) .Proof. Here we prove only ( i ) ⇒ ( v ) ⇒ ( vi ) ⇒ ( i ). The remainder can be found in [17,Theorem 11]. Assume that u is a t -quasi unit, and observe that( λ u ) : ( µ t ) = ( λ u ) : (cid:0) D λ u ( µ t ) (cid:1) = ( λ u ) : ( µ D u t ) = ( λ u ) : ( µ u ) = λµλ + µ u . according to the properties of the parallel sum and the following equalities t : w = D w ( t : w ) = D w t : w (see [8, Lemma 2.3, Lemma 2.4, Proposition 2.7]). Assuming ( v ) it is clear that( λ u ) : t = λ λ u = u : ( λ t ) . Finally, since u ≤ t , property ( vi ) implies that D u t = sup n ∈ N (cid:0) t : ( n u ) (cid:1) = sup n ∈ N (cid:0) ( n t ) : u (cid:1) = D t u = u . (cid:3) Theorem 1.5.
Let t be a form on X and let Y be a linear subspace of X . Then t Y is anextremal point of the convex set { s ∈ F + ( X ) | ≤ s ≤ t } and D t Y t = t Y . Proof.
According to the previous lemma, it is enough to show that t Y is a disjoint part of t . That is, t Y and t − t Y are singular. Let s be a form such that s ≤ t Y and s ≤ t − t Y . Then t Y + s vanishes on Y and t Y + s ≤ t , thus the maximality of t Y implies that s = 0. (cid:3) Z. SEBESTY´EN, ZS. TARCSAY, AND T. TITKOS Bounded positive operators
Let H be a complex Hilbert space with the inner product ( · | · ) and the norm k · k .The set of bounded positive operators will be denoted by B + ( H ). The notation A ≤ B stands for the usual relation ∀ x ∈ H : ( Ax | x ) ≤ ( Bx | x ) . For every A ∈ B + ( H ) we set ∀ x, y ∈ H : t A ( x, y ) := ( Ax | y )which defines a bounded nonnegative form on H . Conversely, in view of the Riesz-representation theorem, the correspondence A t A defines a bijection between boundedpositive operators and bounded nonnegative forms. Consequently, we can define the dom-ination, (strong) absolute continuity, and singularity analogously to the ones defined forforms. We write A ≤ d B if there exists a constant c such that A ≤ c · B . If Bx = 0implies that Ax = 0 for all x ∈ H , we say that A is B -absolutely continuous ( A ≪ B ).The operators A and B are singular ( A ⊥ B ) if 0 is the only positive operator which isdominated by both A and B . Finally, A is strongly B -absolutely continuous ( A ≪ s B ) iffor any sequence ( x n ) n ∈ N ∈ H N (cid:0) ( A ( x n − x m ) | x n − x m ) → ∧ ( Bx n | x n ) → (cid:1) ⇒ ( Ax n | x n ) → . Remark that A ≪ B ⇐⇒ ker B ⊆ ker A and A ⊥ B ⇐⇒ ran A / ∩ ran B / = { } , see [3] or [23]. It was proved by Krein that if M is a closed linear subspace of H and A ∈ B + ( H ), then the set (cid:8) S ∈ B + ( H ) (cid:12)(cid:12) ( S ≤ A ) ∧ (ran S ⊆ M ) (cid:9) possesses a greatest element. This follows immediately from our previous results, andthis is why we say that the form t Y is the short of t to the subspace Y . Indeed, let t ( x, y ) = ( Ax | y ) and consider the form t M ⊥ . Since t M ⊥ is a bounded form, there existsa unique S ∈ B + ( H ) such that t M ⊥ ( x, y ) = ( Sx | y ) and x ∈ M ⊥ = ⇒ t M ⊥ [ x ] = 0 = ⇒ ( Sx | x ) = 0 = ⇒ M ⊥ ⊆ ker S = ⇒ ran S ⊆ M . The maximality of S follows from the maximality of t M ⊥ . Now, since the map A t A isan order preserving positive homogeneous map from B + ( H ) into F + ( H ), the followingtheorem is an immediate consequence of Theorem 1.2. Theorem 2.1.
Let A and B be bounded positive operators on H . Then there is a de-composition of A with respect to B into B -absolutely continuous and B -singular parts.Namely, A = A ≪ ,B + A ⊥ ,B . The decomposition is unique, precisely when A ≪ ,B is dominated by B .Proof. Let A ≪ ,B and A ⊥ ,B be the operators corresponding to ( t A ) ker t B and t A − ( t A ) ker t B ,respectively. (cid:3) Corollary 2.2.
Let B be a bounded positive operator with closed range. Then for every A ∈ B + ( H ) A = A ≪ ,B + A ⊥ ,B . is the unique decomposition of A into B -absolutely continuous and B -singular parts. SHORT-TYPE DECOMPOSITION OF FORMS 7
Proof.
If ran B is closed, then the following two sets are identical according to the well-known theorem of Douglas [5] (cid:8) S ∈ B + ( H ) (cid:12)(cid:12) ( S ≤ A ) ∧ (ran S ⊆ ran B ) (cid:9) = (cid:8) S ∈ B + ( H ) (cid:12)(cid:12) ( S ≤ A ) ∧ ( S ≤ d B ) (cid:9) . Consequently, the uniqueness follows from Theorem 2.1. Since ran B is closed, the inclusionker B ⊆ ker A ≪ ,B implies that ran A ≪ ,B ⊆ ran B . (cid:3) Observe that if ran B is closed, then A ≪ ,B coincides with D B A in the sense of Ando[3], and therefore it is strongly absolutely continuous (or closable) with respect to B .Furthermore, according to [23, Theorem 7] we have the following characterization of closedrange positive operators. Corollary 2.3.
Let B be a bounded positive operator. Then the following are equivalent ( i ) ran B is closed, ( ii ) ∀ A ∈ B + ( H ) : A ≪ ,B ≤ d B , ( iii ) ∀ A ∈ B + ( H ) : D B A ≤ d B .If any of ( i ) − ( iii ) fulfills, then D B A = A ≪ ,B for all A ∈ B + ( H ) . Corollary 2.4.
Let A be a bounded positive operator. Then A ≪ ,B is an extremal point ofthe operator segment [0 , A ] := { S ∈ B + ( H ) | S ≤ A } for all B ∈ B + ( H ) . We remark that the short A M of A to the close linear subspace M of the (complex)Hilbert space H possesses a factorization of the form A M = A / P f M A / , where P f M is defined to be the orthogonal projection onto the subspace f M := A − / h M i ,see Krein [11]. This factorization can hold, of course, only if the underlying space is com-plex. Below we offer an alternative factorization of the operator short that simultaneouslytreats the real and complex cases. In fact, we show that there exists a (real or complex,respectively) Hilbert space H A , associated with the positive operator A , such that A M admits a factorization of the form J A ( I − P ) J ∗ A where J A is the canonical continuous em-bedding of H A into H and P is the orthogonal projection onto an appropriately definedsubspace of H A , associated with M . The construction below is taken from [15].Let us consider the range space ran A , equipped with the inner product ( · | · ) A ∀ x, y ∈ H : ( Ax | Ay ) A = ( Ax | y ) . Note that the operator Schwarz inequality( Ax | Ax ) ≤ k A k ( Ax | x )implies that ( · | · ) A defines an inner product, indeed. Let H A stand for the completion ofthat inner product space. Consider the canonical embedding operator of ran A ⊆ H A into H , defined by ∀ x ∈ H : J A ( Ax ) := Ax.
Then J A is well defined and continuous due to the operator Schwarz inequality above(namely, by norm bound p k A k ). This mapping has a unique norm preserving extensionfrom H A to H which is denoted by J A as well. An easy calculation shows that its adjoint J ∗ A acts as an operator from H to H A possessing the canonical property ∀ x ∈ H : J ∗ A x = Ax.
Z. SEBESTY´EN, ZS. TARCSAY, AND T. TITKOS
This yields the following useful factorization for A : A = J A J ∗ A . Theorem 2.5.
Let H be a Hilbert space and let A ∈ B + ( H ) . For a given subspace M ⊆ H denote by P the orthogonal projection of H A onto the closure of { Ax | x ∈ M } .Then the short of A to M equals J A ( I − P ) J ∗ A .Proof. It is enough to show that the quadratic forms of J A ( I − P ) J ∗ A and t M ⊥ are equal.To verify this let x ∈ H . Then( J A ( I − P ) J ∗ A x | x ) = (( I − P ) Ax | ( I − P ) Ax ) A = dist ( Ax ; ran P )= inf y ∈ M ( Ax − Ay | Ax − Ay ) A = inf y ∈ M ( A ( x − y ) | x − y )= t M ⊥ [ x ] , as it is claimed. (cid:3) The above construction yields another formula for the quadratic form of the shortedoperator:
Corollary 2.6.
Let H be a Hilbert space, A ∈ B + ( H ) and M ⊆ H any closed linearsubspace. Then for any x ∈ H ( J A ( I − P ) J ∗ A x | x ) = ( Ax | x ) − sup {| ( Ax | y ) | | y ∈ M , ( Ay | y ) ≤ } . Proof.
For x ∈ H we have( J A ( I − P ) J ∗ A x | x ) = ( Ax | Ax ) A − ( P ( Ax ) | P ( Ax )) A = ( Ax | x ) − sup {| ( Ax | Ay ) A | | y ∈ M , ( Ay | Ay ) A ≤ } = ( Ax | x ) − sup {| ( Ax | y ) | | y ∈ M , ( Ay | y ) ≤ } , indeed. (cid:3) Corollary 2.7. If A and B are bounded positive operators on the Hilbert space H thenthe quadratic forms of A ≪ ,B and A ⊥ ,B can be calculated by the following formulae: ( A ≪ ,B x | x ) = inf y ∈ ker B ( A ( x − y ) | x − y ) , ( A ⊥ ,B x | x ) = sup {| ( Ax | y ) | | y ∈ ker B, ( Ay | y ) ≤ } . Proof.
Since A ≪ ,B is nothing but the short of A to the closed subspace ker B ⊥ , Theorem2.5 together with the above corollary implies the desired formulae. (cid:3) Additive set functions
In this section we apply our main theorem for finitely additive nonnegative set functions.Our main reference is [16]. We recall first some definitions.Let T be a non-empty set, and let R be a ring of some subsets of T . Let µ and ν be(finitely) additive nonnegative set functions (or charges, for short) on R . We say that ν is strongly absolutely continuous with respect to µ (in symbols ν ≪ s µ ) if for any ε > δ > µ ( R ) < δ implies ν ( R ) < ε for all R ∈ R . It is importantto remark that this notion is referred to as absolutely continuity in [16]. We say that thecharge ν is absolutely continuous with respect to µ if µ ( R ) = 0 implies ν ( R ) = 0 for all R ∈ R . Finally ν and µ are singular if the only charge which is dominated by both ν and µ is the zero charge. SHORT-TYPE DECOMPOSITION OF FORMS 9
Let E be the complex vector space of R -step functions, and define the associated form t ν as follows: ∀ ϕ, ψ ∈ E : t ν ( ϕ, ψ ) := Z T ϕ · ψ d ν. It was proved in [16, Theorem 3.2] that if µ and ν are bounded charges, then ν is stronglyabsolutely continuous with respect to µ if and only if t ν is strongly absolutely continuouswith respect to t µ . Similarly, ν and µ are singular precisely when t ν and t µ are singular.Using this result, the authors proved the classical Lebesgue-Darst decomposition theo-rem. Namely, if µ and ν are bounded charges then the formula ν a : R D t µ t ν [ χ R ]defines a charge on R , such that ν a ≪ s µ and ( ν − ν a ) ⊥ µ . We use this argument below toprovide a ( ≪ , ⊥ )-type decomposition. The following lemma (see [16, Lemma 3.3]) playsan essential role in the proof and may be very useful in deciding the additivity of thecorrespondence R t [ χ R ] for a given form t . Lemma 3.1.
Let T be a non-empty set, and let R be a ring of subsets of T . For a givenform t on E the following statements are equivalent: (i) The set function ϑ : R → R defined by ϑ ( R ) := t [ χ R ] is additive; (ii) t [ ζ ] = t [ | ζ | ] for all ζ ∈ E . The main result of this section is the following short-type decomposition of charges. Herewe emphasize that, in contrast to the Lebesgue-Darst decomposition, this decompositionholds for not necessarily bounded charges as well.
Theorem 3.2.
Let R be a ring of subsets of a non-empty set T , and let µ and ν becharges on R . Then there is a decomposition ν = ν ≪ ,µ + ν ⊥ ,µ , where ν ≪ ,µ ≪ µ and ν ⊥ ,µ ⊥ µ . Furthermore, if ϑ is a charge such that ϑ ≤ ν and ϑ ≪ µ ,then ϑ ≤ ν ≪ ,µ .Proof. Let us define the set function ν ≪ ,µ by ∀ R ∈ R : ν ≪ ,µ ( R ) := ( t ν ) ker t µ [ χ R ] . It is clear that µ ( R ) = 0 implies ν ≪ ,µ ( R ) = 0. Our only claim is therefore to prove theadditivity of ν ≪ ,µ . For this purpose, let ϕ ∈ E . In accordance with the previous lemma, itis enough to show that ( t ν ) ker t µ [ | ϕ | ] = ( t ν ) ker t µ [ ϕ ] . Assume that ϕ = k X i =1 λ i · χ R i , where { λ i } ki =1 are non-zero complex numbers and { R i } ki =1 are pairwise disjoint elementsof R . Define the function ψ as follows ψ := k X i =1 | λ i | λ i · χ Rk + χ T \ S ki =1 Ri . Since | ψ ( t ) | = 1 for all t ∈ T , the multiplication with ψ is a bijection on E . (Note that ψ / ∈ E in general.) As t ν [ ζ ] = t ν [ | ζ | ] for all ζ ∈ E , we have that( t ν ) ker t µ [ ϕ ] = inf ξ ∈E t ν [ ϕ − ξ ] = inf ξ ∈E t ν [ | ϕ − ξ | ]= inf ξ ∈E t ν [ | ψ | · | ϕ − ξ | ] = inf ξ ∈E t ν [ || ϕ | − ψ · ξ | ]= inf ξ ∈E t ν [ | ϕ | − ψ · ξ ] = ( t ν ) ker t µ [ | ϕ | ] . Consequently, ν ≪ ,µ is a charge, which is absolutely continuous with respect to µ . Since ν and ν ≪ ,µ are charges, ν ⊥ ,µ := ν − ν ≪ ,µ is a charge too, which is derived from t ν − ( t ν ) ker t µ .Hence, ν ⊥ ,µ and µ are singular. (cid:3) The following corollary is an immediate consequence of Theorem 1.5.
Corollary 3.3.
Let ν and µ be a charges on R . Then ν ≪ ,µ is an extremal point of theconvex set of those charges that are majorized by ν . Remark 3.4. If R is a σ -algebra, µ and ν are σ -additive (i.e., µ and ν are measures),then the notions of absolute continuity and strong absolute continuity coincide, and hence D t µ t ν = ( t ν ) ker t µ . In this case, the short-type decomposition coincides with the classical Lebesgue decompo-sition. Furthermore, we have the following formula for the absolutely continuous part ∀ R ∈ R : ν ≪ ,µ ( R ) = inf ϕ ∈E Z R | − ϕ ( t ) | d ν ( t ) . If R is an algebra of sets, and we consider finitely additive charges on it, then the involvedabsolute continuity concepts are different. Consequently, there exist µ and ν such that D t µ t ν = ( t ν ) ker t µ . Representable functionals
The Lebesgue-type decomposition of positive functionals were studied by several au-thors, see e.g. [7, 9, 10, 20, 21, 24]. Sz˝ucs in [20] proved that the Lebesgue-type decompo-sition of representable positive functionals can be derived from their induced sesquilinearforms. In this section we present a short-type decomposition for representable positivefunctionals, which corresponds to the short type decomposition of their induced forms.Let A be a complex ∗ -algebra and let f : A → C be a positive linear functional on it(that is, f ( a ∗ a ) ≥ a ∈ A ). The form induced by f will be denoted by t f t f : A × A → C ; ( a, b ) f ( b ∗ a ) . For positive functionals f ≤ g means that t f ≤ t g . The positive functional f is called representable , if there exists a Hilbert space H f , a ∗ -representation π f of A into H f , anda cyclic vector ξ f ∈ H f such that ∀ a ∈ A : f ( a ) = ( π f ( a ) ξ f | ξ f ) f . Such a triple ( H f , π f , ξ f ) is provided by the classical GNS-construction (see [14] for thedetails): namely, denote by N f the set of those elements a such that f ( a ∗ a ) = 0, and let H f stand for the Hilbert space completion of the inner product space (cid:0) A / N f , ( · | · ) f (cid:1) ; ∀ a, b ∈ A : ( a + N f | b + N f ) f := t f ( a, b ) = f ( b ∗ a ) . SHORT-TYPE DECOMPOSITION OF FORMS 11
For a ∈ A let π f ( a ) be the left multiplication by a : ∀ x ∈ A : π f ( a )( x + N f ) := ax + N f . The cyclic vector ξ f is defined as the Riesz-representing vector of the continuous linearfunctional H f ⊇ A / N f → C ; a + N f f ( a ) . Note also that π f ( a ) ξ f = a + N f . We define the absolute continuity and singularity as for forms. Singularity means that thezero functional is the only representable functional which is dominated by both f and g .According to [19, Theorem 2], this is equivalent with the singularity of the forms t f and t g . We say that f is g -absolutely continuous ( f ≪ g ), if ∀ a ∈ A : g ( a ∗ a ) = 0 = ⇒ f ( a ∗ a ) = 0 . A decomposition of f into representable g -absolutely continuous and g -singular parts iscalled short-type decomposition.Now, the short-type decomposition for representable functionals can be stated as fol-lows. Theorem 4.1.
Let f and g be representable positive functionals on the ∗ -algebra A . Then f admits a decomposition f = f ≪ ,g + f ⊥ ,g to a sum of representable functionals, where f ≪ ,g is g -absolutely continuous, f ⊥ ,g and g are singular. Furthermore, f ≪ ,g is the greatest among all of the representable functionals h such that h ≤ f and h ≪ g .Proof. Let M be the following closed subspace of H f M := { a + N f | g ( a ∗ a ) = 0 } and let P be the orthogonal projection from H f onto M . Then M and M ⊥ are π f -invariant subspaces. Since π f is a ∗ -representation, it is enough to prove that M is π f invariant. Let a, x ∈ A and assume that g ( a ∗ a ) = 0. Then π f ( x )( a + N f ) = xa + N f ∈ M because g ( a ∗ x ∗ xa ) = k π g ( x )( a + N f ) k g ≤ k π g ( x ) k g · g ( a ∗ a ) = 0 . Consequently, π f ( x ) h M i ⊆ π f ( x ) h{ a + N f | g ( a ∗ a ) = 0 }i ⊆ M , as it is stated. Now, let us define the functionals f ≪ ,g ( a ) := ( π f ( a )( I − P ) ξ f | ( I − P ) ξ f ) f .f ⊥ ,g ( a ) := ( π f ( a ) P ξ f | P ξ f ) f . Clearly, f ≪ ,g and f ⊥ ,g are representable positive functionals. On the other hand, since M is π f -invariant, we find that f ≪ ,g ( a ∗ a ) = k π f ( a )( I − P ) ξ f k f = k ( I − P ) π f ( a ) ξ f k f = k ( I − P )( a + N f ) k f = t f ≪ ,g [ a ] and similarly, f ⊥ ,g ( a ∗ a ) = k P ( a + N f ) k f = t f ⊥ ,g [ a ] . Since t f ≪ ,g is t g -absolutely continuous, and t f ⊥ ,g is t g -singular, we infer that f ≪ ,g ≪ g and f ⊥ ,g ⊥ g . The maximality of f ≪ ,g follows from the maximality of t f ≪ ,g . (cid:3) Corollary 4.2.
Let f and g be representable positive functionals on the ∗ -algebra A .Then f ≪ ,g is an extremal point of the convex set of those representable functionals thatare majorized by f . References [1] Anderson, Jr., W. N.,
Shorted operators , SIAM J. Appl. Math., 20 (1971), 520–525.[2] Anderson, Jr., W. N. and Trapp, G. E.,
Shorted operators. II. , SIAM J. Appl. Math., 28 (1975),60–71.[3] Ando, T.,
Lebesgue-type decomposition positive operators , Acta. Sci. Math. (Szeged), 38 (1976), 253–260.[4] Darst, R.B.,
A decomposition of finitely additive set functions , J. for Angew. Math., 210 (1962),31–37.[5] Douglas, R.G.,
On majorization, factorization, and range inclusion of operators on Hilbert space ,Proc. Amer. Math. Soc., 17 (1996), 413–416.[6] Eriksson, S.L. and Leutwiler, H.,
A potential theoretic approach to parallel addition , Math. Ann., 274(1986), 301–317.[7] Gudder, S.,
A Radon-Nikodym theorem for ∗ -algebras , Pacific J. Math., 80(1)(1979),141–149.[8] Hassi, S. and Sebesty´en, Z. and de Snoo, H., Lebesgue type decompositions for nonnegative forms , J.Funct. Anal., 257(12) (2009), 3858–3894.[9] Inoue, A.,
A Radon-Nikodym theorem for positive linear functionals on ∗ -algebras , J. Operator The-ory, 10(1983), 77–86.[10] Kosaki, H., Lebesgue decomposition of states on a von Neumann algebra , American Journal of Math.,Vol 107. No.3(1985) 697–735.[11] Krein, M.G.,
The theory of self-adjoint extensions of semi-bounded Hermitian operators , [Mat.Sbornik] 10 (1947), 431-495.[12] Riesz, F.,
Sur quelques notions fondamentales dans la th´eorie g´en´erale des op´erations lin´eaires , Ann.of Math., (2) 41 (1940), 174–206.[13] Rosenberg, M.,
Range decomposition and generalized inverse of nonnegative Hermitian matrices ,SIAM Rev., 11 (1969), 568–571.[14] Sebesty´en, Z.,
On representability of linear functionals on ∗ -algebras , Periodica Math. Hung., 15(3)(1984), 233–239.[15] Sebesty´en, Z., Operator extensions on Hilbert space,
Acta Sci. Math. (Szeged), 57 (1993), 233–248.[16] Sebesty´en, Z. and Tarcsay, Zs. and Titkos, T.,
Lebesgue decomposition theorems , Acta Sci. Math.(Szeged), 79(1-2) (2013), 219–233.[17] Sebesty´en, Z. and Titkos, T.,
Complement of forms , Positivity, 17 (2013), 1–15.[18] Simon, B.,
A canonical decomposition for quadratic forms with applications to monotone convergencetheorems , J. Funct. Anal. 28 (1978), 377–385.[19] Sz˝ucs, Zs.,
The singularity of positive linear functionals , Acta Math. Hung., 136 (1-2) (2012), 138-155.[20] Sz˝ucs, Zs.,
On the Lebesgue decomposition of representable forms over algebras , J. Operator Theory,70:1(2013), 3-31.[21] Sz˝ucs, Zs.,
On the Lebesgue decomposition of positive linear functionals , Proc. Amer. Math. Soc.,141(2013)619–623.[22] Tarcsay, Zs.,
A functional analytic proof of the Lebesgue-Darst decomposition theorem , Real AnalysisExchange, 39(1), 2013/2014, 241–248.[23] Tarcsay, Zs.,
Lebesgue-type decomposition of positive operators , Positivity, Vol. 17(2013), 803–817.[24] Tarcsay, Zs.,
Lebesgue decomposition of representable functional on ∗ -algebras. , (Manuscript).[25] Titkos, T., Lebesgue decomposition of contents via nonnegative forms , Acta Math. Hungar., 140(1-2)(2013), 151-161.[26] Weidmann, J.,
Lineare operatoren in Hilbertr¨aumen , B.G. Teubner, Stuttgart(1976).
SHORT-TYPE DECOMPOSITION OF FORMS 13
Z. Sebesty´en, Zs. Tarcsay, T. Titkos - Institute of Mathematics, E¨otv¨os L. University,P´azm´any P´eter s´et´any 1/c., Budapest H-1117, Hungary;
E-mail address ::