M. D. Voisei, C. Zalinescu
The main focus of this paper is to study multi-valued linear monotone operators in the contexts of locally convex spaces via the use of their Fitzpatrick and Penot functions. Notions such as maximal monotonicity, uniqueness, negative-infimum, and (dual-) representability are studied and criteria are provided.
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Monotone linear relations play important roles in variational inequality problems and quadratic optimizations. In this paper, we give explicit maximally monotone linear subspace extensions of a monotone linear relation in finite dimensional spaces. Examples are provided to illustrate our extensions. Our results generalize a recent result by Crouzeix and Anaya.
Outgoing monotone geodesics of standard subspaces
本文通过构造显式符号，证明了标准子空间中出射单调测地线的正规形式，并区分了来源于Borchers定理的幺正一参数群的测地线与非此类测地线。
In this paper we study, in the relaxed context of locally convex spaces, intrinsic properties of monotone operators needed for the sum conjecture for maximal monotone operators to hold under classical interiority-type domain constraints.
Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop Phelps and sum representability results are presented in the last part.
We generalize the phenomenon of continuation from complex anal- ysis to locally operator monotone functions. Along the lines of the egde-of- the-wedge theorem, we prove continuations exist dependent only on geometric features of the domain and, namely, independent of the function values. We prove a generalization of the Julia inequality for a class of functions containing locally operator monotone functions, Pick functions.
A version of the Dynamical Systems Method for solving ill-posed nonlinear equations with monotone and locally Hölder continuous operators is studied in this paper. A discrepancy principle is proposed and justified under natural and weak assumptions. The only smoothness assumption on $F$ is the local Hölder continuity of order $α>1/2$.
In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on Banach space with locally monotone operators, which is a generalization of the classical result by J.L. Lions for monotone operators. In particular, we show that local monotonicity implies the pseudo-monotonicity. The main result is applied to various types of PDE such as reaction-diffusion equations, generalized Burgers equation, Navier-Stokes equation, 3D Leray-$α$ model and $p$-Laplace equation with non-monotone perturbations.
Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients
本文研究了具有局部单调系数的半线性随机偏微分方程的温和解的存在性和唯一性，利用Nagy扩张定理将问题转化为无限维随机微分方程，并探讨了解的马尔可夫性质。
We show that a complemented subspace of a locally convex direct sum of an uncountable collection of Banach spaces is a locally convex direct sum of complemented subspaces of countable subsums. As a corollary we prove that a complemented subspace of a locally convex direct sum of arbitrary collection of $\ell_{1}(Γ)$-spaces is isomorphic to a locally convex direct sum of $\ell_{1}(Γ)$-spaces.
The local equicontinuity of an operator $T:X\rightrightarrows X^{*}$ with proper Fitzpatrick function $φ_{T}$ and defined in a barreled locally convex space $X$ has been shown to hold on the algebraic interior of $\operatorname*{Pr}\,_{X}(\operatorname*{dom}φ_{T})$). The current note presents direct consequences of the aforementioned result with regard to the local equicontinuity of a maximal monotone operator defined in a barreled locally convex space.
We construct a complete locally convex topological vector space $X$ of countable algebraic dimension and a continuous linear operator $T:X\to X$ such that $T$ has no non-trivial closed invariant subspaces.
Recently in [1] a new class of maximal monotone operators has been introduced. In this note we study domain range properties as well as connections with other classes and calculus rules for these operators we called strongly-representable. While not every maximal monotone operator is strongly-representable, every maximal monotone NI operator is strongly-representable, and every strongly representable operator is locally maximal monotone, maximal monotone locally, and ANA. As a consequence the conjugate of the Fitzpatrick function of a maximal monotone operator is not necessarily a representati…
In this brief note, we show that the hypotheses of Löwner's theorem on matrix monotonicity in several commuting variables as proved by Agler, McCarthy and Young can be significantly relaxed. Specifically, we extend their theorem from continuously differentiable locally matrix monotone functions to arbitrary locally matrix monotone functions using mollification techniques.
Characterizations are given for 1-complemented hyperplanes of strictly monotone real Lorentz spaces and 1-complemented finite codimensional subspaces (which contain at least one basis element) of real Orlicz spaces equipped with either Luxemburg or Orlicz norm.
We prove that for every closed locally convex subspace $E$ of $L_0$ and for any continuous linear operator $T$ from $L_0$ to $L_0/E$ there is a continuous linear operator $S$ from $L_0$ to $L_0$ such that $T = QS$ where $Q$ is the quotient map from $L_0$ to $L_0/E$.
We present a new sufficient condition under which a maximal monotone operator $T:X\tos X^*$ admits a unique maximal monotone extension to the bidual $\widetilde T:X^{**} \rightrightarrows X^*$. For non-linear operators this condition is equivalent to uniqueness of the extension. The class of maximal monotone operators which satisfy this new condition includes class of Gossez type D maximal monotone operators, previously defined and studied by J.-P. Gossez, and all maximal monotone operators of this new class satisfies a restricted version of Brondsted-Rockafellar condition. The central tool in…
Ergodicity and mixing for locally monotone stochastic evolution equations
本文研究了局部单调驱动和退化加性Wiener噪声的随机演化方程，建立了存在唯一不变概率测度的定量条件，并提供了解的矩估计和半群的e-性质，以及2-Wassersteinε-混合时间的上界。
Let E be a locally solid vector lattice. In this paper, we consider two particular vector subspaces of the space of all order bounded operators on E. With the aid of two appropriate topologies, we show that under some conditions, they establish both, locally solid vector lattices and topologically complete topological algebras.
Abstract Equivalent conditions that make the convex subdifferential maximal monotone are investigated in the general settings of locally convex spaces.
A subspace $X$ of a vector space over a field $K$ is hyperinvariant with respect to an endomorphism $f$ of $V$ if it is invariant for all endomorphisms of $V$ that commute with $f$. We assume that $f$ is locally nilpotent, that is, every $ x \in V $ is annihilated by some power of $f$, and that $V$ is an infinite direct sum of $f$-cyclic subspaces. In this note we describe the lattice of hyperinvariant subspaces of $V$. We extend results of Fillmore, Herrero and Longstaff (Linear Algebra Appl. 17 (1977), 125--132) to infinite dimensional spaces.
In this paper we prove that maximal H-monotone operators $T:H^n\rightrightarrows V_1$ whose domain is all the Heisenberg group $H^n$ are locally bounded. This implies that they are upper semicontinuous. As a consequence, maximal H-monotonicity of an operator on $H^n$ can be characterized by a suitable version of Minty's type theorem.
Counter Examples in Non-Archimedean Locally Convex Spaces
本文探讨了非阿基米德局部凸空间中的反例，重点关注非阿基米德Fréchet空间中具有或不具有Schauder基的特殊闭子空间。
It is proved that there exist complemented subspaces of countable topological products (locally convex direct sums) of Banach spaces which cannot be represented as topological products (locally convex direct sums) of Banach spaces. (This is a revised version of the paper by the same name formerly contained in the file ostrovskiicompsbsp.atx.)
This paper is about certain linear subspaces of Banach SN spaces (that is to say Banach spaces which have a symmetric nonexpansive linear map into their dual spaces). We apply our results to monotone linear subspaces of the product of a Banach space and its dual. In this paper, we establish several new results and also give improved proofs of some known ones in both the general and the special contexts.
Uniqueness of Hahn-Banach extensions in locally convex spaces
本文研究局部凸空间中线性泛函的Hahn-Banach延拓唯一性，提出property-SNP并给出其多种刻画，建立其与Banach空间property-U的对应关系，还在逐点收敛拓扑下的连续函数空间上开展相关研究。
We prove generalizations of Löwner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.
Recently it was introduced the so-called Bishop-Phelps-Bollob{á}s property for positive operators between Banach lattices. In this paper we prove that the pair $(C_0(L), Y) $ has the Bishop-Phelps--Bollob{á}s property for positive operators, for any locally compact Hausdorff topological space $L$, whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. In case that the space $C_0(L)$ is separable, the same statement holds for any uniformly monotone Banach lattice $Y .$ We also show the following partial converse of the main result. In case that $Y$ is a strictly monotone Banach l…
In the present work we show that the local generalized monotonicity of a lower semicontinuous set-valued operator on some certain type of dense sets ensures the global generalized monotonicity of that operator. We achieve this goal gradually by showing at first that the lower semicontinuous set-valued functions of one real variable, which are locally generalized monotone on a dense subsets of their domain are globally generalized monotone. Then, these results are extended to the case of set-valued operators on arbitrary Banach spaces. We close this work with a section on the global generalized…
Gelfand-Naimark duality (Commutative $C^*$-algebras $\equiv$ Locally compact Hausdorff spaces) is extended to $C^*$-algebras $\equiv$ Quotient maps on locally compact Hausdorff spaces. Using this duality, we give for an \emph{arbitrary} bounded operator on a complex Hilbert space of several dimensions, a functional calculus and the existence theorem for nontrivial invariant subspace.
Faces of quasidensity
本文研究实Banach空间中对偶乘积的极大单调与拟稠密集，定义Gossez扩展并给出等价刻画，推广满射性定理至非自反情形。
Monotone convex operators and time-consistent systems of operators appear naturally in stochastic optimization and mathematical finance in the context of pricing and risk measurement. We study the dual representation of a monotone convex \emph{operator} when its domain is defined on a subspace of $L_p$, with $p\in [1,\infty]$, and we prove a sandwich preserving extension theorem. These results are then applied to study systems of such operators defined only on subspaces. We propose various dynamic sandwich preserving extension results depending on the nature of time: finite discrete, countable…
Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex …
Let G be a locally compact group. In this note, we characterise non-degenerate *-representations of A_Φ(G) and B_Φ(G). We also study spectral subspaces associated to a non-degenerate Banach space representation of A_Φ(G).
We define the finest order on inductive limits of ordered cones which makes the linear mappings monotone and gives rise to the definition of inductive limit topologies for cones. Using the polars of neighborhoods, we establish embeddings between direct sums, inductive limits and their duals. These lead us to investigate the weak topologies and the topologies of pointwise convergence in inductive limits.
We prove that if $X$ is a complex strictly monotone sequence space with $1$-unconditional basis, $Y \subseteq X$ has no bands isometric to $\ell_2^2$ and $Y$ is the range of norm-one projection from $X$, then $Y$ is a closed linear span a family of mutually disjoint vectors in $X$. We completely characterize $1$-complemented subspaces and norm-one projections in complex spaces $\ell_p(\ell_q)$ for $1 \leq p, q < \infty$. Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are $1$-complemented in (real or complex) Orlicz or …
Complemented subspaces of a $C(K)$-space constructed by Candido
本文研究了由Candido构造的非平凡局部紧致Hausdorff空间$L$所对应的$C_0(L \times L)$空间的补子空间分类问题。
We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a Köthe function space $E$ is upper locally uniformly monotone if and only if its $p$-conv…
Subspace Dual and orthogonal frames\\ by action of an abelian group
本文利用 Zak 变换研究局部紧群中阿贝尔子群平移标架的子空间对偶与正交性，并推广到样条、Gabor 系统及局部紧阿贝尔群等情形。
Given a function $f: (a,b) \rightarrow \mathbb{R},$ Löwner's theorem states $f$ is monotone when extended to self-adjoint matrices via the functional calculus, if and only if $f$ extends to a self-map of the complex upper half plane. In recent years, several generalizations of Löwner's theorem have been proven in several variables. We use the relaxed Agler, McCarthy and Young theorem on locally matrix monotone functions in several commuting variables to generalize results in the noncommutative case. Specifically, we show that a real free function defined over an operator system must analytical…
Let $M$ be a Hopf--von Neuman algebra with the predual $M_*$ and $WAP(M)$ the subspace in $M$ composed of weakly almost periodic functionals on $M_*$. The main example of such an algebra is $M=L^\infty(\mathbb G)$ for a locally compact quantum group $\mathbb G$. We define a pair of left/right spaces $WAP_{iso,l}(M)$ and $WAP_{iso,r}(M)$ inside $WAP(M)$ and prove that they carry invariant means. These spaces are currently the widest known to admit invariant means in the quantum setting. In the case when $M=L^\infty(G)$ and $G$ is a locally compact group, these spaces are equal to $WAP(G)$.
Given a linear semi-bounded symmetric operator $S\ge -ω$, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators $A_Θ$ of type $λ>ω$ (i.e. generators of one-parameter continuous nonlinear semi-groups of contractions of type $λ$) which coincide with the Friedrichs extension of $S$ on a convex set containing ${\mathscr D}(S)$. The extension parameter $Θ\subset{\mathfrak h}\times{\mathfrak h}$ ranges over the set of nonlinear maximal monotone relations on an auxiliary Hilbert space $\mathfrak h$ isomorphic to the deficiency subspace of $S$. Moreover $A_…
Given a function in the Hardy space of inner harmonic gradients on the sphere, H+(S), we consider the problem of finding a corresponding function in the Hardy space of outer harmonic gradients on the sphere, H-(S), such that the sum of both functions differs from a locally supported vector field only by a tangential divergence-free contribution. We characterize the subspace of H+(S) that allows such a continuation and show that it is dense but not closed within H+(S). Furthermore, we derive the linear mapping that maps a vector field from this subspace of H+(S) to the corresponding unique vect…
We study geometric properties of the Banach space $\mathcal{R}$ constructed recently by C.~Read (arXiv 1307.7958) which does not contain proximinal subspaces of finite codimension greater than or equal to two. Concretely, we show that the bidual of $\mathcal{R}$ is strictly convex, that the norm of the dual of $\mathcal{R}$ is rough, and that $\mathcal{R}$ is weakly locally uniformly rotund (but it is not locally uniformly rotund). Apart of the own interest of the results, they provide a simplification of the proof by M.~Rmoutil (J.\ Funct.\ Anal.\ 272 (2017), 918--928) that the set of norm-at…
In this note we prove that the reproducing kernel of a Hilbert space satisfying the division property has integrable form, is locally of trace class, and the Hilbert space itself is a Hilbert space of holomorphic functions.
In this paper, we construct new important the subspaces $D_{p}^{q}S$, $D_{p}^{q}W$, $D_{p}^{q}F$ and $D_{p}^{q}B$ for a locally convex FK-space $X$ containing $ϕ$, the space of finite sequences. Then we show that there is relation among these subspaces. Also, we study deferred Cesàro conullity of one FK-space with respect to another, and we give some important results. Finally, we examine the deferred Cesàro conullity of the absolute summability domain $l_A$, and show that if $l_A$ is deferred Cesàro conull, then $A$ cannot be $l$-replaceable.
We introduce the strong Gelfand-Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand-Phillips property among locally convex spaces admitting a stronger Banach space topology. If $C_{\mathcal T}(X)$ is a space of continuous functions on a Tychonoff space $X$, endowed with a locally convex topology $\mathcal T$ between the pointwise topology and the compact-open topology, then: (a) the space $C_{\mathcal T}(X)$ has the strong Gelfand-Phillips property iff $X$ contains a compact countable subspace $K\subseteq X$ of fin…
We characterize 1-complemented subspaces of finite codimension in strictly monotone one-$p$-convex, $2<p<\infty,$ sequence spaces. Next we describe, up to isometric isomorphism, all possible types of 1-unconditional structures in sequence spaces with few surjective isometries. We also give a new example of a class of real sequence spaces with few surjective isometries.
The bracket map was originally considered for locally compact abelian groups. In this work we extend the study of bracket maps to the noncommutative setting, providing characterizations of bases and frames for cyclic subspaces of the Heisenberg group. We also indicate how to generalize these results to a class of non-abelian nilpotent Lie groups whose irreducible representations are square integrable modulo the center.
We study Banach spaces satisfying some geometric or structural properties involving tightness of transfinite sequences of nested linear subspaces. These properties are much weaker than WCG and closely related to Corson's property (C). Given a transfinite sequence of normalized vectors, which is dispersed or null in some sense, we extract a subsequence which is a biorthogonal sequence, or even a weakly null monotone basic sequence, depending on the setting. The Separable Complementation Property is established for spaces with an M-basis under rather weak geometric properties. We also consider a…
In this paper, we study the noncommutative Orlicz space $L_φ(\widetilde{\mathcal{M}},τ)$, which generalizes the concept of noncommutative $L^{p}$ space, where $\mathcal{M}$ is a von Neumann algebra, and $φ$ is an Orlicz function. As a modular space, the space $L_φ(\widetilde{\mathcal{M}},τ)$ possesses the Fatou property, and consequently, it is a Banach space. In addition, a new description of the subspace $E_φ(\widetilde{\mathcal{M}},τ)=\overline{\mathcal{M}\bigcap L_φ(\widetilde{\mathcal{M}},τ)}$ in $L_φ(\widetilde{\mathcal{M}},τ)$, which is closed under the norm topology and dense under the…
Gelfand duality between unital commutative C*-algebras and Compact Hausdorff spaces is extended to all unital C*-algebras, where the dual objects are what we call compact Hausdorff quantum spaces. We apply this result to obtain, a characterization of unitary groups of C*-algebras, and, for arbitrary bounded Hilbert space operators, (i) A spectral theorem cum continuous functional calculus, and (ii) A proof of the general Invariant Subspace Theorem. Also described is a nonabelian generalization of Pontryagin duality of abelian locally compact groups.
Lower semicontinuity of monotone functionals in the mixed topology on $C_b$
本文刻画 $C_b$ 上单调泛函的从下连续性与混合拓扑下半连续性的等价关系，并由此推导凸单调映射的可数可加测度对偶表示、Choquet 积分及等度连续性结论。
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