Continuous Linear Map Research Articles

Nonlinear maps preserving asymptotic equivalence on B(X)

Let X be an infinite-dimensional complex Banach space. By B(X) we denote the algebra of all bounded linear operators on X. It is shown that if Φ : B(X) → B(X) is a surjective map satisfying A + B is asymptotically equivalent to C if and only if Φ(A) + Φ(B) is asymptotically equivalent to Φ(C), then either there exist bijective continuous linear or conjugate-linear maps T, S : X → X such that Φ(A) = TAS for every A ∈ B(X), or there exist bijective continuous linear or conjugate-linear maps T : X* → X and S : X → X * such that Φ(A) = TA* S for every A ∈ B(X).

CTIFTrack: Continuous Temporal Information Fusion for object track

In visual tracking tasks, researchers usually focus on increasing the complexity of the model or only discretely focusing on the changes in the object itself to achieve accurate recognition and tracking of the moving object. However, they often overlook the significant contribution of video-level linear temporal information fusion and continuous spatiotemporal mapping to tracking tasks. This oversight may lead to poor tracking performance or insufficient real-time ability of the model in complex scenes. Therefore, this paper proposes a real-time tracker, namely Continuous Temporal Information Fusion Tracker (CTIFTrack). The key of CTIFTrack lies in its well-designed Temporal Information Fusion (TIF) module, which cleverly performs a linear fusion of the temporal information between the (t-1)-th and the t-th frames and completes the spatiotemporal mapping. This enables the tracker to better understand the overall spatiotemporal information and contextual spatiotemporal correlations within the video, thereby having a positive impact on the tracking task. In addition, this paper also proposes the Object Template Feature Refinement (OTFR) module, which effectively captures the global information and local details of the object, and further improves the tracker’s understanding of the object features. Extensive experiments are conducted on seven benchmarks, such as LaSOT, GOT-10K, UAV123, NFS, TrackingNet, VOT2018 and OTB-100. The experimental results validate the significant contribution of the TIF module and OTFR module to the tracking task, as well as the effectiveness of CTIFTrack. It is worth noting that while maintaining excellent tracking performance, CTIFTrack also shows outstanding real-time tracking speed. On the Nvidia Tesla T4-16GB GPU, the FPS of CTIFTrack reaches 71.98. The code and demo materials will be available at https://github.com/vpsg-research/CTIFTrack.

A Really Simple Elementary Proof of the Uniform Boundedness Principle in F -Spaces

Abstract We give a proof of the uniform boundedness principle for linear continuous maps from F -spaces into topological vector spaces which is elementary and also quite simple.

About the image of strongly generalized derivations of order $n$

Let A and B be two algebras. A linear mapping 𝜟:A → B is called a strongly generalizedderivation of order n, if there exist the families {𝐸_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐹_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐺_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} 𝑎𝑛𝑑 {𝐻_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} of linear mappings which satisfy 𝛥(𝑎𝑏) = Σ [𝐸_𝑘(𝑎) 𝐹_𝑘(𝑏) + 𝐺_𝑘(𝑎)𝐻_𝑘(𝑏)] 𝑛𝑘 =1 for all 𝑎, 𝑏 ⋲ 𝐴.The main purpose of this paper is to study the image of such derivations. Our main result onthe image of strongly generalized derivations of order one reads as follows: Let A be a unital,commutative Banach algebra and let 𝛥: 𝐴 → 𝐴 be a continuous strongly generalizedderivation of order one; that is, there exist the linear mappings 𝐸, 𝐹, 𝐺, 𝐻: 𝐴 → 𝐴 satisfying𝐷(𝑎𝑏) = 𝐸(𝑎) 𝐹(𝑏) + 𝐺(𝑎) 𝐻(𝑏) for all 𝑎, 𝑏 ⋲ 𝐴. Let 𝐸, 𝐹, 𝐺 and 𝐻 be continuous linearmappings. We prove that, under certain conditions, 𝐻 (𝐴), 𝐸(𝐴) 𝑎𝑛𝑑 𝛥(𝐴) are contained inthe Jacobson radical of A. This result generalizes Singer-Wermer theorem about the image ofcontinuous derivations on commutative Banach algebras.

Pairs of continuous linear bijective maps preserving fixed products of operators

Let X X be a complex Banach space, and denote by B ( X ) \mathcal {B}(X) the algebra of all bounded linear operators on X X . Let C , D ∈ B ( X ) C,D\in \mathcal {B} \left ( X\right ) be fixed operators. In this paper, we characterize linear, continuous and bijective maps φ \varphi and ψ \psi on B ( X ) \mathcal {B}\left ( X\right ) for which there exist invertible operators T 0 , W 0 ∈ B ( X ) T_0, W_0 \in \mathcal { B}(X) such that φ ( T 0 ) , ψ ( W 0 ) ∈ B ( X ) \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that φ ( A ) ψ ( B ) = D \varphi \left ( A\right ) \psi \left ( B\right ) =D in B ( X ) \mathcal {B}(X) whenever A B = C AB=C in B ( X ) \mathcal {B}(X) . As a corollary, we deduce the form of linear, bijective and continuous maps φ \varphi on B ( X ) \mathcal {B}(X) having the property that φ ( A ) φ ( B ) = D \varphi \left ( A\right ) \varphi \left ( B\right ) =D in B ( X ) \mathcal {B}(X) whenever A B = C AB=C .

Some classes of topological spaces extending the class of Δ-spaces

A study of the class Δ \Delta consisting of topological Δ \Delta -spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes Δ 2 ⊂ Δ 1 \Delta _2 \subset \Delta _1 properly containing Δ \Delta . We observe that for every first-countable X X the following equivalences hold: X ∈ Δ 1 X\in \Delta _1 iff X ∈ Δ 2 X\in \Delta _2 iff each countable subset of X X is G δ G_{\delta } . Thus, new proposed concepts provide a natural extension of the family of all λ \lambda -sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X X belongs to the class Δ 1 \Delta _1 iff countable subsets of X X are scattered. (2) Every regular scattered space belongs to the class Δ 2 \Delta _2 . We investigate whether the classes Δ 1 \Delta _1 and Δ 2 \Delta _2 are invariant under the basic topological operations. Similarly to Δ \Delta , both classes Δ 1 \Delta _1 and Δ 2 \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to Δ \Delta , they are not preserved by closed continuous images. Let Y Y be l l -dominated by X X , i.e. C p ( X ) C_p(X) admits a continuous linear map onto C p ( Y ) C_p(Y) . We show that Y ∈ Δ 1 Y \in \Delta _1 whenever X ∈ Δ 1 X \in \Delta _1 . Moreover, we establish that if Y Y is l l -dominated by a compact scattered space X X , then Y Y is a pseudocompact space such that its Stone–Čech compactification β Y \beta Y is scattered.

Locally linear attributes of ReLU neural networks.

A ReLU neural network functions as a continuous piecewise linear map from an input space to an output space. The weights in the neural network determine a partitioning of the input space into convex polytopes, where each polytope is associated with a distinct affine mapping. The structure of this partitioning, together with the affine map attached to each polytope, can be analyzed to investigate the behavior of the associated neural network. We investigate simple problems to build intuition on how these regions act and both how they can potentially be reduced in number and how similar structures occur across different networks. To validate these intuitions, we apply them to networks trained on MNIST to demonstrate similarity between those networks and the potential for them to be reduced in complexity.

Integrating geometries of ReLU feedforward neural networks.

This paper investigates the integration of multiple geometries present within a ReLU-based neural network. A ReLU neural network determines a piecewise affine linear continuous map, M, from an input space ℝm to an output space ℝn. The piecewise behavior corresponds to a polyhedral decomposition of ℝm. Each polyhedron in the decomposition can be labeled with a binary vector (whose length equals the number of ReLU nodes in the network) and with an affine linear function (which agrees with M when restricted to points in the polyhedron). We develop a toolbox that calculates the binary vector for a polyhedra containing a given data point with respect to a given ReLU FFNN. We utilize this binary vector to derive bounding facets for the corresponding polyhedron, extraction of "active" bits within the binary vector, enumeration of neighboring binary vectors, and visualization of the polyhedral decomposition (Python code is available at https://github.com/cglrtrgy/GoL_Toolbox). Polyhedra in the polyhedral decomposition of ℝm are neighbors if they share a facet. Binary vectors for neighboring polyhedra differ in exactly 1 bit. Using the toolbox, we analyze the Hamming distance between the binary vectors for polyhedra containing points from adversarial/nonadversarial datasets revealing distinct geometric properties. A bisection method is employed to identify sample points with a Hamming distance of 1 along the shortest Euclidean distance path, facilitating the analysis of local geometric interplay between Euclidean geometry and the polyhedral decomposition along the path. Additionally, we study the distribution of Chebyshev centers and related radii across different polyhedra, shedding light on the polyhedral shape, size, clustering, and aiding in the understanding of decision boundaries.

On the derivations of the quadratic Jordan product in the space of rectangular matrices

Let Mn,m be a rectangular finite dimensional Cartan factor, i.e. the space L(Cn,Cm) with 1≤n≤m, and let δ:Mn,m→Mn,m be a quadratic Jordan derivation of Mn,m, i.e., a map (neither linearity nor continuity of δ is assumed) that satisfies the functional equationδ{ABA}={δ(A)BA}+{Aδ(B)A}+{ABδ(A)},(A,B∈Mn,m), where (A,B,C)↦{AB,C}:=12(AB⁎C+CB⁎A) stands for the Jordan triple product in Mn,m. We prove that then δ automatically is a continuous complex linear map on Mn,m. More precisely we show that δ admits a representation of the form δ(A)=UA+AV, (A∈Mn,m), for a suitable pair U,V of square skew symmetric matrices with complex entries U∈Mn,n and V∈Mm,m.

Lozi map embedded into the 2D border collision normal form

A 2D piecewise linear continuous two-parameter map known as the Lozi map is a special case of the 2D border collision normal form depending on four parameters. In the present paper, we investigate how the bifurcation structure of the Lozi map is incorporated into the bifurcation structure of the 2D border collision normal form using an analytical representation of the boundaries of the largest periodicity regions related to the cycles with rotation number 1/n, . At the centre bifurcation boundary of the stability domain of the fixed point both maps are conservative which leads to a quite intricate bifurcation structure near this boundary.

Maps on $ C^\ast $-algebras are skew Lie triple derivations or homomorphisms at one point

<abstract><p>In this paper, we show that every continuous linear map between unital $ C^\ast $-algebras is skew Lie triple derivable at the identity is a $ \ast $-derivation and that every continuous linear map between unital $ C^\ast $-algebras which is a skew Lie triple homomorphism at the identity is a Jordan $ \ast $-homomorphism.</p></abstract>

Linear orthogonality preservers between function spaces associated with commutative JB⋆-triples

It is known, by Gelfand theory, that every commutative JB∗-triple admits a representation as a space of continuous functions of the formC0T(L)={a∈C0(L):a(λt)=λa(t),∀λ∈T,t∈L},where L is a principal T-bundle and T denotes the unit circle in C. We provide a full technical description of all orthogonality preserving (non-necessarily continuous nor bijective) linear maps between commutative JB∗-triples. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB∗-triples is automatically continuous and bi-orthogonality preserving.

Maps preserving two-sided zero products on Banach algebras

Let A and B be Banach algebras with bounded approximate identities and let Φ:A→B be a surjective continuous linear map which preserves two-sided zero products (i.e., Φ(a)Φ(b)=Φ(b)Φ(a)=0 whenever ab=ba=0). We show that Φ is a weighted Jordan homomorphism provided that A is zero product determined and weakly amenable. These conditions are in particular fulfilled when A is the group algebra L1(G) with G any locally compact group. We also study a more general type of continuous linear maps Φ:A→B that satisfy Φ(a)Φ(b)+Φ(b)Φ(a)=0 whenever ab=ba=0. We show in particular that if Φ is surjective and A is a C⁎-algebra, then Φ is a weighted Jordan homomorphism.

Jordan *-triple derivations on the exceptional Cartan factors

Let U be any of the two exceptional Cartan factors V and VI and let δ:U→U be a Jordan *-triple derivation of U, that is, a map (neither linearity nor continuity of δ is assumed) that satisfies the functional equation δ{ab⁎a}={δ(a)b⁎a}+{aδ(b)⁎a}+{ab⁎δ(a)}, (a,b∈U), where (a,b)↦{ab⁎a} stands for the Jordan triple product in U. We give an explicit representation of δ as certain multipliers on U and prove that δ automatically is a continuous real linear map on U. This gives a new description of the real Banach Lie algebra of Jordan triple derivations of U.

On linear continuous operators between distinguished spaces $$C_p(X)$$

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), for a Tychonoff space X, a locally convex space $$C_{p}(X)$$ is distinguished if and only if X is a $$\Delta $$ -space. If there exists a linear continuous surjective mapping $$T:C_p(X) \rightarrow C_p(Y)$$ and $$C_p(X)$$ is distinguished, then $$C_p(Y)$$ also is distinguished (Ka̧kol and Leiderman Proc AMS Ser B, 2021). Firstly, in this paper we explore the following question: Under which conditions the operator $$T:C_p(X) \rightarrow C_p(Y)$$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $$C_p([1,\alpha ])$$ , where $$\alpha $$ is a countable ordinal number. A complete characterization of all Y which admit a linear continuous surjective mapping $$T:C_p([1,\alpha ]) \rightarrow C_p(Y)$$ is given. We also observe that for every countable ordinal $$\alpha $$ all closed linear subspaces of $$C_p([1,\alpha ])$$ are distinguished, thereby answering an open question posed in Ka̧kol and Leiderman (Proc AMS Ser B, 2021). Using some properties of $$\Delta $$ -spaces we prove that a linear continuous surjection $$T:C_p(X) \rightarrow C_k(X)_w$$ , where $$C_k(X)_w$$ denotes the Banach space C(X) endowed with its weak topology, does not exist for every infinite metrizable compact C-space X (in particular, for every infinite compact $$X \subset {\mathbb {R}}^n$$ ).

Estimates of the topological degree of a class of piecewise linear maps with applications

We provide some new estimates for the topological degree of a class of continuous and piecewise linear maps based on a classical integral computation formula. We provide applications to some nonlinear problems that exhibit a local [Formula: see text] structure.

Hyperreflexivity of the space of module homomorphisms between non-commutative Lp-spaces

Let M be a von Neumann algebra, and let 0<p,q≤∞. Then the space HomM(Lp(M),Lq(M)) of all right M-module homomorphisms from Lp(M) to Lq(M) is a reflexive subspace of the space of all continuous linear maps from Lp(M) to Lq(M). Further, the space HomM(Lp(M),Lq(M)) is hyperreflexive in each of the following cases: (i) 1≤q<p≤∞; (ii) 1≤p,q≤∞ and M is injective, in which case the hyperreflexivity constant is at most 8.

Identities related to generalized derivations and jordan (*,*)-derivations

The main purpose of this research is to characterize generalized (left) derivations and Jordan (*,*)-derivations on Banach algebras and rings using some functional identities. Let A be a unital semiprime Banach algebra and let F,G : A ? A be linear mappings satisfying F(x) =-x2G(x-1) for all x ? Inv(A), where Inv(A) denotes the set of all invertible elements of A. Then both F and G are generalized derivations on A. Another result in this regard is as follows. Let A be a unital semiprime algebra and let n &gt; 1 be an integer. Let f,g : A ? A be linear mappings satisfying f (an) = nan-1g(a) = ng(a)an-1 for all a ? A. If g(e) ? Z(A), then f and g are generalized derivations associated with the same derivation on A. In particular, if A is a unital semisimple Banach algebra, then both f and 1 are continuous linear mappings. Moreover, we define a (*,*)-ring and a Jordan (*,*)-derivation. A characterization of Jordan (*,*)-derivations is presented as follows. Let R be an n!-torsion free (*,*)-ring, let n &gt; 1 be an integer and let d : R ? R be an additive mapping satisfying d(an) = ?nj =1 a*n-jd(a)a* j-1 for all a ? R. Then d is a Jordan (*,*)-derivation on R. Some other functional identities are also investigated.

Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

Pseudodifferential operators involving linear canonical Hankel transformations on some ultradifferentiable function spaces

In this work, continuity of linear canonical Hankel transformation on the ultradifferentiable function spaces are discussed. Furthermore, the pseudodifferential operator associated with linear canonical Hankel transformation is shown to be the continuous linear mapping from one ultradifferentiable function space to another. An application to a Dirichlet problem is discussed.