Automatic Boundedness of Some Operators Between Ordered and Topological Vector Spaces

Order-to-topology continuous and order-to-norm bounded operators have been studied recently by different authors [1–7]. In the present note, some of their results are extended to the setting of topological vector spaces and new conditions for automatic boundedness of operators are given. We abbreviate normed, topological, ordered, ordered normed, and ordered Banach vector spaces as NS, TVS, OVS, ONS, and OBS, respectively. In what follows, vector spaces are real, operators are linear, symbol L ( X,Y ) stands for the space of operators from a vector space X to a vector space Y, B x for the closed unit ball of a NS X , and x _a | 0 for a decreasing net in an OVS such that inf _a x _a = 0. A net ( x _a ) in an OVS X

• - order converges to x ( o -converges to x , or x _a —> x ) if there exists a net g e | 0 in X such that, for each в there is a e such that ± ( x _a — x ) C g e for a ^ a e ;

• ^# The work was carried out in the framework of the State Task to the Sobolev Institute of Mathematics, project no. FWNF-2026-0022.

(0 2026 Emelyanov, E. Yu.

– relative uniform converges to x ( ru -converges to x , or x _α - ^r → ^u x ) if, for some u ∈ X + there exists an increasing sequence ( a n ) of indices with ± ( x _a — x ) C n u for a ^ a n .

We need the following classes of operators.

Definition 1.1. An operator T between OVSs X and Y is

• a )    order bounded if T [ a, b ] is order bounded for every order interval [ a, b ] in X (the set of such operators will be denoted by Lob ( X, Y )).

• b )    ru-continuous if Tx _a —— 0 in Y whenever x _a —— 0 (shortly, T G L _rc ( X,Y )).

An operator T from OVS X to TVS ( Y, т ) is

• c )    order-to-topology continuous if Tx _a — 0 whenever x _a — 0 (shortly, T G L _OT c ( X,Y )).

If Y is a NS, the set of such operators is denoted by L _onc ( X, Y ).

• d )    ru-to-topology continuous if Tx _a — 0 whenever x _a — r — 0 (shortly, T G L _rT c ( X,Y )).

If Y is a NS, the set of such operators is denoted by L _rnc ( X, Y ).

• e )    order-to-topology bounded if T [ a, b ] is т -bounded for every [ a, b ] С X (shortly, T G Lo _T b ( X, Y )). If Y is a NS, the set of such operators is denoted by L _on b ( X, Y ).

Clearly, all sets of operators mentioned in Definition 1.1 are vector spaces. We also recall several definitions concerning collective convergences and collectively qualified sets of operators (see, [2-4; 8-10]). For convenience, we say that a family B = { ( x a ) _aeA } beB of nets in a TVS X = ( X, т ) indexed by the same directed set A collective т-converges to 0 (briefly, B —— 0) whenever, for each U G т (0) there exists a u with x^b _a G U for all a ^ a u and b G B .

Definition 1.2. A set T of operators between OVSs X and Y is

• a )    collectively order bounded ( T G L ob ( X,Y )), if T [ a, b ] is order bounded for every [ a, b ] С X .

A set T of operators from an OVS X to a TVS ( Y, т ) is

• b )    collectively order-to-topology continuous ( T G L _OT c ( X,Y )), if T x _a c —— 0 whenever x _a — 0. If additionally Y is a NS, we write T G L _onc ( X, Y ).

• c ) collectively ru-to-topology continuous ( T G L _rT c ( X,Y )), if T x _a c ^T> 0 whenever

x _a - r — 0. If additionally Y is a NS, we write T G L _rnc ( X, Y ).

• d )    collectively order-to-topology bounded ( T G L o _T b ( X,Y )), if T [ a, b ] is т -bounded for every [ a,b ] С X . If additionally Y is a NS, we write T G L _on b ( X,Y ).

A set T of operators from a TVS ( X, £ ) to a TVS ( Y, т ) is

• e )    collectively continuous ( T G L _c ( X,Y )), if T x _a -c—— 0 whenever x _a — 0.

• f )    collectively bounded ( T G L b ( X,Y )), if T U is т -bounded whenever U is ^ -bounded.

• 2. Main Results

The σ -versions of Definitions 1.1 and 1.2 are obtained via replacing nets by sequences, and the corresponding spaces (classes) are denoted by L0 _TC ( X,Y ) ( L 0 _T c ( X,Y )), etc. Let X and Y be OVSs. Then r i T + r 2 T 2 , T i U T G L ob ( X, Y ) for every r i ,r 2 G R and nonempty subsets T 1 , T 2 of L ob ( X, Y ). The same is true for L o_TC ( X, Y ), L _rT c ( X,Y ), L ^ _TC( X,Y ), etc.

The present note is organized as follows. Theorem 2.1 asserts collective ru-to-topology continuity of collectively order-to-topology bounded sets. In Theorem 2.2, conditions for the inclusion L o_TC ( X, Y ) С L o _T b ( X,Y ) are given. Theorem 2.3 gives conditions for the inclusion L oTb ( X,Y ) С L b ( X,Y ).

For the terminology and notations that are not explained in the text, we refer to [11, 12].

We start with the following theorem which tells us that collectively order-to-topology bounded sets quite often agree with collectively ru-to-topology continuous sets.

Theorem 2.1. Let X be an OVS and ( Y,t ) a TVS. Then L o_Tb ( X,Y ) C L _rTc ( X,Y ) . If additionally X . is generating then L o _T b ( X, Y ) = L _rT c ( X, Y ) .

• < 1 If, on the contrary, T E L_{O T} b ( X, Y ) \ L _rT c ( X, Y ) then, for some x _a —e 0 there exists an absorbing U E т (0) such that, for every a there exist a' ^ a and T _a E T with T _a x _a ‘ / U .

Since x _a —e 0, for some u E X . there exists an increasing sequence ( a n ) of indices with ±nx _a C u for a ^ a n . It follows from T E L o_Tb ( X, Y ) that T [ — u, u ] C NU for some N E N. Since nx _a E [ —u,u ] for a ^ a n and [ a n ] ' ^ a n then nx [ _a n] ’ E [ —u,u ] , and hence T _a n ( nx [ _a n y ) E NU for every n . In particular, T _a N ( x [ _a N ] ‘ ) E U , which is absurd. We conclude L oTb ( X,Y ) C L rτc ( X,Y ).

Now, suppose X ₊ is generating, and let T E L _rT c ( X, Y ) \ L o_Tb ( X, Y ). Since X = X ₊ — X ₊ and T / L_{O T}b( X,Y ), there exist x E X ₊ and an absorbing U E т (0) with T [ — x,x ] C nU for every n E N. Therefore, T n x n / nU for all n and some sequences ( x _n ) in [ —x, x ] and ( T n ) in T . Since - x _n —e 0 and T E L _rTC ( X,Y ), there exists a sequence ( n k ) such that T ( - x n ) E U for all n ^ n k and T E T . Then T _n 1 x _n 1 E n - U , which is a contradiction. So, L rTc ( X, Y ) C L o?b ( X, -Y ), and hence L oTb (^X, -Y ) — L rrc ( X, -Y ). ^

Corollary 2.1. Every order-to-topology bounded operator from an OVS X to a TVS ( Y, т ) is ru-to-topology continuous. Assuming in addition that X . is generating, L _OT b ( X, Y ) — LrTc ( X,Y ) .

The following result can be viewed as a topological version of [4, Theorem 2.1] (in the vector lattice setting see also [2, Theorem 2.1]), [11, Lemma 1.72] and [13, Theorem 2.1].

Theorem 2.2. Let X be an Archimedean OVS with a generating cone and ( Y,t ) a TVS. Then L otc ( X,Y ) C L oTb ( X,Y ) .

• <    We argue to a contradiction supposing T E L _o>TC ( X,Y ) \ L _oT b ( X,Y ). Theorem 2.1 implies T / L _rTC ( X,Y ). So, T x _a — ^T> 0 for some net ( x _a ) in X such that x _a —e 0. Since X is Archimedean then x _a —> 0, and hence T x _a ^c Tt 0 because of T E L _oiTC( X,Y ). The obtained contradiction completes the proof. >

Corollary 2.2. Every order-to-topology continuous operator from an Archimedean OVS with a generating cone to a TVS is order-to-topology bounded.

It is well known that L _c ( X,Y ) — L b ( X, Y ) whenever X and Y are NSs. The following result can be viewed as a partial extension of [2, Theorem 2.1] and [4, Theorem 2.8].

Theorem 2.3. Let X be an OBS with a closed generating cone and ( Y,t ) a TVS. Then L oTb ( X,Y ) C L b ( X,Y ) .

• <    Let T E L _oT b( X,Y ). Suppose, on the contrary, T / L b ( X, Y ). By the Krein-Smulian theorem (cf. [12, Theorem 2.37]), aB x C B x П X ₊ — B x П X ₊ for some a >  0, and hence T ( B x A X + ) is not т -bounded. Then, there exists an absorbing U E т (0) with T ( B x A X ₊ ) C nU for every n E N. So, for some sequences ( x n ) in B x П X ₊ and ( T n ) in T we have T _n x _n / n ³ U for all n . Set

x :— IP || — £ n-^ E X ₊ .

n =1

Since T E L _oT b( X,Y ) then T [0 , x ] C NU for some N E N. It follows from n ^{- 2} x n E [0 ,x ] that T n ( n ^-2 x n ) E NU C nU for large enough n . This is absurd, because T n ( n ^-2 x n ) / nU for all n . >

Corollary 2.3. Every collectively order-to-norm bounded set of operators from an OBS with a closed generating cone to a NS is uniformly bounded.

Corollary 2.4. Every order-to-norm bounded operator from an OBS with a closed generating cone to a NS is bounded.

The following corollary of Theorem 2.3 provides an automatic continuity result that extends the well known fact (see, for example, [12, Theorem 2.32]) that every positive operator from an OBS with a closes generating cone to an OBS with a closed cone is continuous.

Corollary 2.5. Every order bounded operator from an OBS with a closed generating cone to an ONS with a normal cone is continuous.

Since in NSs bounded linear operators are continuous, and since weak compact sets are bounded in Banach spaces, we obtain the following consequence of Theorem 2.3.

Corollary 2.6. Every operator from an OBS with a closed generating cone to a Banach space is continuous whenever it takes order intervals onto relatively weak compact sets.

Collective boundedness of a semigroup generated by a single operator is known as power boundedness of the operator. We say that an operator T on an ordered TVS ( X, т ) is power order-to-topology bounded if the set U^ =i T n [ a,b ] is т -bounded for every [ a, b ] in X . An operator semigroup S on an ordered TVS ( X, т ) is order-to-topology bounded if the set Ut e S T [ a,b ] is т -bounded for every [ a,b ] in X . The next two corollaries deal with these notions.

Corollary 2.7. Every power order-to-norm bounded operator on an OBS with a closed generating cone is power bounded.

Corollary 2.8. Every power order-to-norm bounded operator semigroup on an OBS with a closed generating cone is uniformly continuous.

Since L b ( X, Y ) C L onb ( X, Y ) whenever X is a normal OVS and Y is a NS, the next collective extension of [3, Proposition 1.5] follows from Theorems 2.1 and 2.3.

Proposition 2.1. Let X be an OBS with a closed generating normal cone and Y a NS. Then Lm c ( X, Y ) = L onb ( X, Y ) = L b ( X, Y ) .

Corollary 2.9. Let X be an OBS with a closed generating normal cone and Y a NS. Then Lrnc ( X, Y ) = Lo _n b ( X, Y ) = Lb ( X, Y ) .

By [4, Theorem 2.4], L _rc ( X, Y ) = Lob ( X, Y ) whenever X and Y are OVSs with generating cones. Therefore, we have one more consequence of Proposition 2.1 that provides conditions for automatic boundedness of ru-continuous operators.

Corollary 2.10. Let X be an OBS with a closed generating normal cone and Y an OVS with a generating normal cone. Then L _rc ( X, Y ) C Lb ( X, Y ) .

Proposition 2.2 [4, Proposition 2.11] . Let X and Y be OBSs with closed generating cones, and X ~ = 0 . Then every T G L ob ( X,Y ) is uniformly bounded iff Y . is normal.

• < 1 If Y ₊ is not normal, then the interval [O ,y o ] is not bounded for some y o G X ^ . Pick f o = 0 in X ~ and take x o G X with f o ( x o ) = 1. Then, { f o 0 y } _{y e}[0 ,_{y 0}] G L_o b ( X, Y ). However, the set { f o 0 y } y _e[o ,y ₀] is not uniformly bounded since

U ^{( f} o ⁰ y ⁾⁽ x o ) = ^{[0 ,}y o ^]

ye [o ,y o ]

is not bounded.

If Y ₊ is normal, then L ob ( X, Y ) C L _onb ( X, Y ) by [4, Proposition 2.6]. The rest follows from Proposition 2.1. >

The following proposition is an extension of [3, Lemma 2.1].

Proposition 2.3. Let X be a normal OBS and ( Y, т ) a Banach space with a dual topology. Then L° _C ( X, Y ) C Lonb ( X, Y ) .

• < 1 On the way to contradiction, assume T G L0 _TC ( X, Y ) \ L _onb ( X, Y ) Then T [0 , u ] is not bounded for some u G X ₊ . Since X is normal, [0 ,u ] is bounded, say sup _{x e}[0 ,u ] ||x || C M . Pick a sequence ( u _n ) in [0 , u ] with | Tu _n | ^ n 2 n , and set

∞

У п ^:= II • II - ^ 2 - k U k for n G N. k = n

Then y _n ^ ^ 0. Let 0 С y o C y n for all n G N. Since 0 С У о C 2 ^{1 -n} u and ||2 ^{1 -n} u | C M 2 ^{1- n} G 0 then y o =0 by [12, Theorem 2.23]. So, y _n ^ 0, and hence y n —> 0. We deduce from T G L 0T_C ( X, Y ) that Ty n G 0. Since the topology т is dual then Ty n —G 0. Therefore, the sequence ( T n y n ) is norm bounded. This is absurd because

|| Ty n +i - Ty n || = || T ( 2 ^-n u n )|| >  n ( V n G N) . >

Corollary 2.11. Every order-to-topology σ -continuous operator from a normal OBS to a Banach space with a dual topology is order-to-norm bounded.

Combining Corollaries 2.11 and 2.4, gives the last result of our note.

Corollary 2.12. Every order-to-topology σ -continuous operator from an OBS with a closed generating normal cone to a Banach space with a dual topology is bounded.