In this short article we show an orthogonal decomposition of a Hilbert space as a sum of null solutions of the first derivative and the first derivative of a traceless higher order Hilbert/Sobolev space. We define orthogonal projections and see some of its properties and show case some decomposition of elementary functions as corollaries.
Contrary to the simple structure of the tensor product of the quaternionic Hilbert space, the octonionic situation becomes more involved. It turns out that an octonionic Hilbert space can be decomposed as an orthogonal direct sum of two subspaces, each of them isomorphic to a tensor product of an irreducible octonionic Hilbert space with a real Hilbert space. As an application, we find that for a given orthogonal basis the octonionic Parseval equality holds if and only if the basis is weak associative. Fortunately, there always exists a weak associative orthogonal basis in an octonionic Hilber…
In the first part of the paper, we use states on $C^*$-algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^*$-module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^*$-module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^*$-module under the assumption that their inner product has negative real part. We introduce the concept of Pythagoras orth…
In this paper, the concept of Birkhoff--James orthogonality of operators on a Hilbert space is generalized when a semi-inner product is considered. More precisely, for linear operators $T$ and $S$ on a complex Hilbert space $\mathcal{H}$, a new relation $T\perp^B_A S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying ${\|T + γS\|}_A\geq {\|T\|}_A$ for all $γ\in \mathbb{C}$. We extend a theorem due to R. Bhatia and P. Šemrl, by proving that $T\perp^B_A S$ if and only if there exists a sequence of $A$-unit vectors $\{x_n\}$ in $\math…
Let $\big(\mathscr{X}, \langle\cdot, \cdot\rangle\big)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathscr{A}$ and let $\mathcal{S}(\mathscr{A})$ be the set of states on $\mathscr{A}$. In this paper, we first compute the norm derivative for elements $x$ and $y$ of $\mathscr{X}$ as follows \begin{align*} ρ_{_{+}}(x, y) = \max\Big\{\mbox{Re}\,φ(\langle x, y\rangle): \, φ\in \mathcal{S}(\mathscr{A}), φ(\langle x, x\rangle) = \|x\|^2\Big\}. \end{align*} We then apply it to characterize different concepts of orthogonality in $\mathscr{X}$. In particular, we present a simpler proof of the clas…
The decomposition into interaction subspaces is an important result for graphical models and plays a central role for results on the linearized marginal problem; similarly the Chaos decomposition plays an important role in statistical physics at the thermodynamic limit and in probability theory in general. We unify and extend both constructions by defining and characterizing decomposable functors from a well-founded poset to the category of Hilbert spaces, completing previous work on decomposable collections of vector subspaces and presheaves.
Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $$ \mathbb E Ψ(N_t) \leq C_{Φ,Ψ,X} \mathbb E Φ(M_t),\;\;\; t\geq 0, $$ where $Φ, Ψ:X \to \mathbb R_+$ are convex continuous functions and the least admissible constant $C_{Φ,Ψ,X}$ coincides with the $Φ,Ψ$-norm of the periodic Hilbert transform. As a corollary, it is shown that the $Φ,Ψ$-norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transf…
We classify tuples of (not necessarily commuting) isometries that admit von Neumann-Wold decomposition. We introduce the notion of twisted isometries for tuples of isometries and prove the existence of orthogonal decomposition for such tuples. The former classification is partially inspired by a result that was observed more than three decades ago by Gaspar and Suciu. And the latter result generalizes Popovici's orthogonal decompositions for pairs of commuting isometries to general tuples of twisted isometries which also includes the case of tuples of commuting isometries. Our results unify al…
Let $H$ be a complex Hilbert space and let ${\mathcal G}_{k}(H)$ be the Grassmannian formed by $k$-dimensional subspaces of $H$. Suppose that $\dim H>2k$ and $f$ is an orthogonality preserving injective transformation of ${\mathcal G}_{k}(H)$, i.e. for any orthogonal $X,Y\in {\mathcal G}_{k}(H)$ the images $f(X),f(Y)$ are orthogonal. If $\dim H=n$ is finite, then $n=mk+i$ for some integers $m\ge 2$ and $i\in \{0,1,\dots,k-1\}$ (for $i=0$ we have $m\ge 3$). We show that $f$ is a bijection induced by a unitary or anti-unitary operator if $i\in \{0,1,2,3\}$ or $m\ge i+1\ge 5$; in particular, the …
University of South Florida（南佛罗里达大学）；Farmingdale State College（法明戴尔州立学院）；Changshu Institute of Technology（常熟理工学院）
This manuscript gives a theoretical framework for a new Hilbert space of functions, the so called occupation kernel Hilbert space (OKHS), that operate on collections of signals rather than real or complex numbers. To support this new definition, an explicit class of OKHSs is given through the consideration of a reproducing kernel Hilbert space (RKHS). This space enables the definition of nonlocal operators, such as fractional order Liouville operators, as well as spectral decomposition methods for corresponding fractional order dynamical systems. In this manuscript, a fractional order DMD rout…
Let $n$ be any natural number. The $n$-centered operator is introduced for adjointable operators on Hilbert $C^*$-modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, $n$-centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator $T$, if $T$ is Moore-Penrose invertible and is $n$-centered, then its Moore-Penrose inverse is also $n$-centered. A Hilbert space operator $T$ is constructed such that $T…
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributions in order to implement the kernel Bayes rule and build sequential data models. It was recently shown that transfer operators such as the Perron-Frobenius or Koopman operator can also be approximated in a similar fashion using covariance and cross-covariance operators and that …
