Nilpotent Operators Research Articles

Structure of k‐Quasi‐(m, n)‐Isosymmetric Operators

The investigation of new operators belonging to some specific classes has been quite fashionable since the beginning of the century, and sometimes it is indeed relevant. In this study, we introduce and study a new class of operators called k‐quasi‐(m, n)‐isosymmetric operators on Hilbert spaces. This new class of operators emerges as a generalization of the (m, n)‐isosymmetric operators. We give a characterization for any operator to be k‐quasi‐(m, n)‐isosymmetric operator. Using this characterization, we prove that any power of an k‐quasi‐(m, n)‐isosymmetric operator is also an k‐quasi‐(m, n)‐isosymmetric operator. Furthermore, we study the perturbation of an k‐quasi‐(m, n)‐isosymmetric operator with a nilpotent operator. The product and tensor products of two k‐quasi‐(m, n)‐isosymmetries are investigated.

29. MAPPING ASSOCIATIONS BETWEEN ATTENTION-DEFICIT/HYPERACTIVITY DISORDER GENETIC LIABILITY, ATTENTION-DEFICIT/HYPERACTIVITY DISORDER SYMPTOMS, AND BEHAVIORAL AND NEURAL CORRELATES OF RESPONSE INHIBITION

We prove the existence of Hall polynomials for x2-bounded invariant subspaces of nilpotent linear operators.

Local commutants and ultrainvariant subspaces

For an operator A on a complex Banach space X and a closed subspace M⊆X, the local commutant of A at M is the set C(A;M) of all operators T on X such that TAx=ATx for every x∈M. It is clear that C(A;M) is a closed linear space of operators, however it is not an algebra, in general. For a given A, we show that C(A;M) is an algebra if and only if the largest subspace MA of X such that C(A;M)=C(A;MA) is invariant for every operator in C(A;M). We say that a closed subspace M⊆X is ultrainvariant for A∈B(X) if it is invariant for every operator in C(A;M). For several types of operators we prove that they have non-trivial ultrainvariant subspaces. For a normal operator on a Hilbert space, every hyperinvariant subspace is ultrainvariant. On the other hand, the family of all ultrainvariant subspaces of a nilpotent operator can be a proper subset of the lattice of all hyperinvariant subspaces.

Drazin invertibility, characterizations and structure of polynomially normal operators

The class of polynomially normal operators is a wider class than the class of all normal operators. Inspired by some interesting well known facts about normal operators and by some recent work, we present new properties of polynomially normal operators. Precisely, we prove that under certain conditions polynomially normal operators are Drazin or even group invertible and we also give necessary and sufficient conditions for a polynomially normal operator to have a closed range. In addition, we characterize polynomially normal operators with or without closed ranges applying the adequate operator matrix representations. Furthermore, we show that in some cases a polynomially normal operator can be written as a direct sum of a normal operator and a nilpotent operator. What is more, we state several examples to illustrate our results.

H1 ≠ h1 for Anderson t-motives

Let M be an Anderson t-motive of dimension n and rank r. Associated are two Fq[T]-modules H1(M), H1(M) of dimensions h1(M), h1(M)≤r - analogs of H1(A,Z), H1(A,Z) for an abelian variety A. There is a theorem (Anderson): h1(M)=r⇔h1(M)=r; in this case M is called uniformizable. It is natural to expect that always h1(M)=h1(M). Nevertheless, we explicitly construct a counterexample. Further, we answer a question of D. Goss: is it possible that two Anderson t-motives that differ only by a nilpotent operator N are of different uniformizability type, i.e. one of them is uniformizable and other not? We give an explicit example that this is possible. Finally, explicit formulas for calculation of h1(M), h1(M) obtained in the present paper will be used in future for systematic calculation of h1, h1 of all Anderson t-motives. Moreover, the first step of this calculation (for a class of t-motives) is already made in a forthcoming paper of the authors.

Squashing activation functions in benchmark tests: Towards a more eXplainable Artificial Intelligence using continuous-valued logic

Over the past few years, deep neural networks have shown excellent results in multiple tasks, however, there is still an increasing need to address the problem of interpretability to improve model transparency, performance, and safety. Logical reasoning is a vital aspect of human intelligence. However, traditional symbolic reasoning methods are mostly based on hard rules, which may only have limited generalization capability. Achieving eXplainable Artificial Intelligence (XAI) by combining neural networks with soft, continuous-valued logic and multi-criteria decision-making tools is one of the most promising ways to approach this problem: by this combination, the black-box nature of neural models can be reduced. The continuous logic-based neural model uses so-called Squashing activation functions, a parametric family of functions that satisfy natural invariance requirements and contain rectified linear units as a particular case. This work demonstrates the first benchmark tests that measure the performance of Squashing functions in neural networks. Three experiments were carried out to examine their usability and a comparison with the most popular activation functions was made for five different network types. The performance was determined by measuring the accuracy, loss, and time per epoch. These experiments and the conducted benchmarks have proven that the use of Squashing functions is possible and similar in performance to conventional activation functions. Moreover, a further experiment was conducted by implementing nilpotent logical gates to demonstrate how simple classification tasks can be solved successfully and with high performance. The results indicate that due to the embedded nilpotent logical operators and the differentiability of the Squashing function, it is possible to solve classification problems, where other commonly used activation functions fail.

Composite and Background Fields in Non-Abelian Gauge Models

A joint introduction of composite and background fields into non-Abelian quantum gauge theories is suggested based on the symmetries of the generating functional of Green’s functions, with the systematic analysis focused on quantum Yang–Mills theories, including the properties of the generating functional of vertex Green’s functions (effective action). For the effective action in such theories, gauge dependence is found in terms of a nilpotent operator with composite and background fields, and on-shell independence from gauge fixing is established. The basic concept of a joint introduction of composite and background fields into non-Abelian gauge theories is extended to the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion, as well as to the Gribov–Zwanziger theory.

Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix

I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A,B) of commuting nilpotent operators on a vector space contains the problem of classifying an arbitrary t-tuple of linear operators. Moreover, it contains the problem of classifying representations of an arbitrary quiver and an arbitrary finite-dimensional algebra, and so it is considered as hopeless.If (A,B) is such a pair, then KerA∩KerB≠0. We give a simple normal form (Anor,Bnor) of the matrices of (A,B) if KerA∩KerB is one-dimensional. We do not know whether it is canonical; i.e., whether (Anor,Bnor) is uniquely determined by (A,B). We prove its uniqueness only if the Jordan canonical form of A is a direct sum of Jordan blocks of the same size and the field is of zero characteristic.The matrix Anor is the Weyr canonical form of A, and Bnor commutes with Anor. In order to describe the structure of (Anor,Bnor), we describe explicitly all matrices commuting with a given Weyr matrix.

On evolution operators in characteristic 2

We are interested in the evolution operators defined on commutative and non-associative algebras when the characteristic of the scalar field is 2. We distinguish four types: nilpotent, quasi-constant, ultimately periodic, and plenary train operators. They are studied and classified for non-baric and for baric algebras.

Invariant subspaces for Bishop operators and beyond

Bishop operators Tα acting on L2[0,1) were proposed by E. Bishop in the fifties as possible operators which might entail counterexamples for the Invariant Subspace Problem. We prove that all the Bishop operators are biquasitriangular and, derive as a consequence that they are norm limits of nilpotent operators. Moreover, by means of arithmetical techniques along with a theorem of Atzmon, the set of irrationals α∈(0,1) for which Tα is known to possess non-trivial closed invariant subspaces is considerably enlarged, extending previous results by Davie [11], MacDonald [21] and Flattot [14]. Furthermore, we essentially show that when our approach fails to produce invariant subspaces it is actually because Atzmon's Theorem cannot be applied. Finally, upon applying arithmetical bounds obtained, we deduce local spectral properties of Bishop operators proving, in particular, that neither of them satisfy Dunford's property (C).

Some results on higher orders quasi-isometries

The purpose of the present paper is to pursue further study of a class of linear bounded operators, known as $n$-quasi-$m$-isometric operators acting on an infinite complex separable Hilbert space ${\mathcal H}$. We give an equivalent condition for any $T$ to be $n$-quasi-$m$-isometric operator. Using this result we prove that any power of an $n$-quasi-$m$-isometric operator is also an $n$-quasi-$m$-isometric operator. In general the converse is not true. However, we prove that if $T^r$ and $T^{r+1}$ are $n$-quasi-$m$-isometries for a positive integer $r$, then T is an $n$-quasi-$m$-isometric operator. We study the sum of an $n$-quasi-$m$-isometric operator with a nilpotent operator. We also study the product and tensor product of two $n$-quasi-$m$-isometries. Further, we define $n$-quasi strict $m$-isometric operators and prove their basic properties.

Combinatorial Algorithms for Binary Operations on LR-tableaux with Entries Equal to 1 with Applications to Nilpotent Linear Operators

In the paper we investigate an algorithmic associative binary operation * on the set ℒℛ1 of Littlewood-Richardson tableaux with entries equal to one. We extend * to an algorithmic nonassociative binary operation on the set ℒℛ1 × ℕ and show that it is equivalent to the operation of taking the generi c extensions of objects in the category of homomorphisms from semisimple nilpotent linear operators to nilpotent linear operators. Thus we get a combinatorial algorithm computing generic extensions in this category.

The Equations Defining Affine Grassmannians in Type A and a Conjecture of Kreiman, Lakshmibai, Magyar, and Weyman

AbstractThe affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called shuffles, and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps; first, we reinterpret the shuffle equations in terms of Frobenius twists of symmetric functions. Using this, we reduce to a finite-dimensional problem, which we solve. For the 2nd step, we introduce a finite-dimensional analogue of the affine Grassmannian of $SL_n$, which we conjecture to be precisely the reduced subscheme of a finite-dimensional Grassmannian consisting of subspaces invariant under a nilpotent operator.

Formal Functional Calculus for Weakly Locally Nilpotent Operators in Fréchet Spaces

We generalize the classical Riesz–Dunford holomorphic functional calculus with weakly locally nilpotent operators in a Frechet space to the case of the algebra of all formal power series. We obtain a counterpart of the spectral mapping theorem and formulas similar to the Taylor and Riesz–Dunford formulas. We show that the weak local nilpotency of an operator is necessary for constructing the formal functional calculus. Bibliography: 12 titles.

The Distance from a Rank Projection to the Nilpotent Operators on

AbstractBuilding on MacDonald’s formula for the distance from a rank-one projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$ is $\frac {1}{2}\sec (\frac {\pi }{\frac {n}{n-1}+2} )$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where Q is a projection of rank $n-1$ and $T\in \mathbb {M}_n(\mathbb {C})$ is a nilpotent of minimal distance to Q. Furthermore, we prove that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.

Some Examples of m-Isometries

We obtain the admissible sets on the unit circle to be the spectrum of a strict $m$-isometry on an $n$-finite dimensional Hilbert space. This property gives a better picture of the correct spectrum of an $m$-isometry. We determine that the only $m$-isometries on $\mathbb{R}^2$ are $3$-isometries and isometries giving by $\pm I+Q$, where $Q$ is a nilpotent operator. Moreover, on real Hilbert space, we obtain that $m$-isometries preserve volumes. Also we present a way to construct a strict $(m+1)$-isometry with an $m$-isometry given, using ideas of Aleman and Suciu \cite[Proposition 5.2]{AS} on infinite dimensional Hilbert space.

Brane webs and magnetic quivers for SQCD

It is widely considered that the classical Higgs branch of 4d mathcal{N} = 2 SQCD is a well understood object. However there is no satisfactory understanding of its structure. There are two complications: (1) the Higgs branch chiral ring contains nilpotent elements, as can easily be checked in the case of SU(N) with 1 flavour. (2) the Higgs branch as a geometric space can in general be decomposed into two cones with nontrivial intersection, the baryonic and mesonic branches. To study the second point in detail we use the recently developed tool of magnetic quivers for five-brane webs, using the fact that the classical Higgs branch for theories with 8 supercharges does not change through dimensional reduction. We compare this approach with the computation of the hyper-Kähler quotient using Hilbert series techniques, finding perfect agreement if nilpotent operators are eliminated by the computation of a so called radical. We study the nature of the nilpotent operators and give conjectures for the Hilbert series of the full Higgs branch, giving new insights into the vacuum structure of 4d mathcal{N} = 2 SQCD. In addition we demonstrate the power of the magnetic quiver technique, as it allows us to identify the decomposition into cones, and provides us with the global symmetries of the theory, as a simple alternative to the techniques that were used to date.

A characterization of the relation between two $\ell $-modular correspondences

Let F be a non archimedean local field of residual characteristic p and ℓ a prime number different from p. Let V denote Vignéras’ ℓ-modular local Langlands correspondence [7], between irreducible ℓ-modular representations of GL n (F) and n-dimensional ℓ-modular Deligne representations of the Weil group W F . In [4], enlarging the space of Galois parameters to Deligne representations with non necessarily nilpotent operators allowed us to propose a modification of the correspondence of Vignéras into a correspondence C, compatible with the formation of local constants in the generic case. In this note, following a remark of Alberto Mínguez, we characterize the modification C∘V -1 by a short list of natural properties.

Semi-generalized Partial Isometries

The purpose of this paper is to present in Hilbert spaces some results for a new class of operators called semi-generalized partial isometries which include two important classes in operator theory: partial isometries and nilpotent operators. We prove some basic properties and decomposition theorems. Some spectral theorems for this class of operators are also given. Part of the results proved in this paper improve and generalize some results known for partial isometries.

On n-quasi-[m,C]-isometric operators

For positive integers m and n, an operator T in B ( H ) is said to be an n-quasi-[m,C]-isometric operator if there exists some conjugation C such that . In this paper, some basic structural properties of n-quasi-[m,C]-isometric operators are established with the help of operator matrix representation. As an application, we obtain that a power of an n-quasi-[m,C]-isometric operator is again an n-quasi-[m,C]-isometric operator. Moreover, we show that the class of n-quasi-[m,C]-isometric operators is norm closed. Finally, we examine the stability of n-quasi-[m,C]-isometric operator under perturbation by nilpotent operators commuting with T.