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A truncated Toeplitz operator is the compression $A_{\phi}:\K_{\Theta} \to \K_{\Theta}$ of a Toeplitz operator $T_{\phi}:H^2\to H^2$ to a model space $\K_{\Theta} := H^2 \ominus \Theta H^2$. For $\Theta$ inner, let $\T_{\Theta}$ denote the…
Cartesian harmonic tensors are completely symmetric traceless tensors in three dimensional space constructed from the direct product of unit vectors. They are useful in generating expressions for the angular coupling of spherical harmonics…
Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special…
For a symmetric pair $(G,H)$ of reductive groups we extend to a large class of generalized principal series representations our previous construction of meromorphic families of symmetry breaking operators. These operators intertwine between…
This article presents a natural extension of the tensor algebra. In addition to "left multiplications" by vectors, we can consider "derivations" by covectors as basic operators on this extended algebra. These two types of operators satisfy…
Following a general method proposed earlier, we construct here Wigner functions defined on coadjoint orbits of a class of semidirect product groups. The groups in question are such that their unitary duals consist purely of representations…
We build the general conformally invariant linear wave operator for a free, symmetric, second-rank tensor field in a d-dimensional ($d\geqslant 2$) metric manifold, and explicit the special case of maximally symmetric spaces. Under the…
We study invariant operators in general tensor models. We show that representation theory provides an efficient framework to count and classify invariants in tensor models. In continuation and completion of our earlier work, we present two…
Higher-order tensors appear in various areas of mechanics as well as physics, medicine or earth sciences. As these tensors are highly complex, most are not well understood. Thus, the analysis and the visualization process form a highly…
In this paper, we mainly focus on how to generalize some conclusions from nonnegative irreducible tensors to nonnegative weakly irreducible tensors. To do so, a basic and important lemma is proven using new tools. First, we give the…
In this paper, we define and study the arithmetic of the ring of $\mathbb{U}$-operators for reductive $p$-adic groups. These operators generalise the notion of "successor" operators for trees with a marked end. We show that they are…
By the use of complete orthonormal sets of nonrelativistic scalar orbitals introduced by the author in previous papers the new complete orthonormal basis sets for two- and four-component spinor wave functions, and Slater spinor orbitals…
While monotone operator theory is often studied on Hilbert spaces, many interesting problems in machine learning and optimization arise naturally in finite-dimensional vector spaces endowed with non-Euclidean norms, such as…
In this article we give general neccessary and sufficient conditions to ensure that a pseudo-Riemannian manifold is conformal to an Einstein space. These conditions are algorithmic in \emph{the metric tensor} whenever the Weyl endomorphism…
General classical theories of material fields in an arbitrary Riemann-Cartan space are considered. For these theories, with the help of equations of balance, new non-trivially generalized, manifestly generally covariant expressions for…
We investigate an extension of Schauder's theorem by studying the relationship between various $s$-numbers of an operator $T$ and its adjoint $T^*$. We have three main results. First, we present a new proof that the approximation number of…
Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces such that their intersection contains a fixed vector space D. It is of interest to make a precise…
The subdifferential of convex functions of the singular spectrum of real matrices has been widely studied in matrix analysis, optimization and automatic control theory. Convex optimization over spaces of tensors is now gaining much interest…
A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose(adjoint). A structure theorem for invertible skew-symmetric operators, which is analogous to the finite…
In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a…