Nilpotent Operators Research Articles

Some Results about Acts over Monoid and Bounded Linear Operators

الأثر V بالنسبة إلى sinshT و خواصه قد تم دراسته في هذا البحث حيث تم دراسة علاقة الأثر المخلص والاثر المنتهى التولد والاثر المنفصل وربطها بالمؤثرات المتباينة حيث تم بهنة العلاقات التالية ان الاثر اذا وفقط اذا مقاس في حالة كون المؤثر هو عديم القوة وكذلك في حالة كون المؤثر شامل فان الاثر هو منتهي التولد اي ان الغضاء هو منتهي التولد وايضا تم برهن ان الاثر مخلص لكل مؤثر مقيد وك\لك قد تم التحقق من انه لاي مؤثر مقيد فان الاثر منفصل وفي حالة كون الاثر منفصل فان المؤثر سوف يكون متباين وايضا تم برهنة انه في حالة كون الفضاء يمتلك قاعدة فان الاثر سوف يكون كل عنصر فيه يمكن كتابته كتركيبة خطية ومن العلاقات المهمة التي قد تم برهانها انه اذا كان المؤثر مشابه لمؤثر اخر فان الاثر سوف يكون نوثيرين بوجد شرط على الفضاء .

A stronger version of Dixmier's averaging theorem and some applications

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given finite elements X1,⋯,Xn∈M, there is a finite decomposition of the identity into integer N∈N mutually orthogonal nonzero projections Ej∈M, I=∑j=1NEj, such that EjXiEj=τ(Xi)Ej for all j=1,⋯,N and i=1,⋯,n. Equivalently, there is a unitary operator U∈M such that 1N∑j=0N−1U⁎jXiUj=τ(Xi)I for i=1,⋯,n. This result is a stronger version of Dixmier's averaging theorem for type II1 factors. As the first application, we show that all elements of trace zero in a type II1 factor are single commutators and any self-adjoint elements of trace zero are single self-commutators. This result answers affirmatively Question 1.1 in [6]. As the second application, we prove that any self-adjoint element in a type II1 factor can be written a linear combination of 4 projections. This result answers affirmatively Question 6(2) in [12]. As the third application, we show that if (M,τ) is a finite factor, X∈M, then there exists a normal operator N∈M and a nilpotent operator K such that X=N+K.

Discrete global symmetries: gauging and twisted compactification

Discrete global symmetries of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs are studied via two operations: gauging and twisted compactification. We consider gauging of discrete symmetries in several well-known 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs, including SU(n) SQCD with 2n flavors, theories of class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} of type A2n−1, and Argyres-Douglas theories of type (AN, AN), as well as propose new 4d SCFTs as a result. The wreathing technique, which involves gauging a subgroup of the automorphism group of the quiver diagram of the corresponding 3d mirror theory, is exploited. This allows us to understand several properties of discretely gauged theories, including moduli spaces and how discrete gauging affects the mixed ’t Hooft anomaly between the 1-form symmetry and the 0-form flavor symmetry. Many examples are viewed through the lens of the Argyres-Seiberg duality and its generalization. We also examine discrete gauging of SU(2) SQCD with 4 flavors by various ℤ2 and ℤ2 × ℤ2 subgroups of the permutation group S4 using the superconformal index. Regarding compactification, we propose a magnetic quiver for 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SU(n) SQCD with 2n flavors compactified on a circle with a ℤ2 twist. The twisted compactification by non-invertible symmetries of the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM theory with gauge group SU(N) is revisited. The non-invertible symmetry naturally gives rise to a ℤk action on the scalar fields parametrizing the moduli space. Upon examining the ℤk invariant chiral ring of the Higgs branch, we find that, in addition to the largest branch of the moduli space that is expected to be captured by the ABJ(M) theory, there exist in general nilpotent operators that lead to a branch of the moduli space which is a radical ideal.

On k-Quasi-(m, n, C)-Isosymmetric Operators

We study the set of k-quasi-(m, n,C)-isosymmetric operators. This family extends the
 set of (m, , n,C)-isosymmetric operators. In the present article, we give operator matrix
 representation of k-quasi-(m, , n,C)-isosymmetric operator in order to obtain some structural
 properties for such operators. we show that if R is k-quasi-(m, n,C)-isosymmetric,
 then Rq is a k-quasi-(m, n,C)-isosymmetric operator. We show that the product of an
 k1-quasi-(m1, n1,C)-isosymmetric and an n2-quasi-(m2, n2,C)-isosymmetric which are Cdouble
 commuting is an max{k1, k2}-quasi-(m1 + m2 − 1, n1 + n2 − 1,C)-isosymmetry
 under suitable conditions. In particular, we prove the stability of perturbation of kquasi-(
 m, n,C)-isosymmetric operator by a nilpotent operator of order p under suitable
 conditions. Moreover, we give some results about the joint approximate spectrum of a
 k-quasi-(m, n,C)-isosymmetric operators.

A note on a generalized Jordan form of an infinite upper triangular matrix

In this paper, several equivalent conditions for the existence of a generalized Jordan form for matrices of locally nilpotent linear operators acting on an infinite countable dimensional vector space are presented, and an example is given showing that these conditions are not always fulfilled, which means the existence of infinite upper triangular matrices that are not similar to any generalized infinite Jordan matrix. Besides, it is shown that these conditions are not applicable in the case of an arbitrary dimensional of a vector space.

Equivariant K-theory of Springer varieties

The aim of this paper is to describe the topological equivariant K-ring, in terms of generators and relations, of a Springer variety Fλ of type A associated to a nilpotent operator having Jordan canonical form whose block sizes form a weakly decreasing sequence λ=(λ1,…,λl). This parallels the description of the equivariant cohomology ring of Fλ due to Abe and Horiguchi and generalizes the description of ordinary topological K-ring of Fλ due to Sankaran and Uma [13].

Gauge-Invariant Lagrangian Formulations for Mixed-Symmetry Higher-Spin Bosonic Fields in AdS Spaces

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux Y(s1,…,sk) with k≥2 rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a k=2 Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic sp(2k) algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the su(1,1) algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for Y(s1,s2) is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin (s1,s2) is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin (1,1) is obtained.

A tale of N cones

We study particular families of bad 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 quiver gauge theories, whose Higgs branches consist of many cones. We show the role of a novel brane configuration in realizing the Higgs moduli for each distinct cone. Through brane constructions, magnetic quivers, Hasse diagrams, and Hilbert series computations we study the intricate structure of the classical Higgs branches. These Higgs branches are both non-normal (since they consist of multiple cones) and non-reduced (due to the presence of nilpotent operators in the chiral ring). Applying the principle of inversion to the classical Higgs branch Hasse diagrams, we conjecture the quantum Coulomb branch Hasse diagrams. These Coulomb branches have several most singular loci, corresponding to the several cones in the Higgs branch. We propose the Hasse diagrams of the full quantum moduli spaces of our theories. The quivers we study can be taken to be 5d effective gauge theories living on brane webs. Their infinite coupling theories have Higgs branches which also consist of multiple cones. Some of these cones have decorated magnetic quivers, whose 3d Coulomb branches remain elusive.

N-Quasi-m-Complex Symmetric Transformations

Our aim in this study is to consider a generalization of the concept of m-complex symmetric transformations to n-quasi-m-complex symmetric transformations. A map S∈B(Y) is said to be an n-quasi-m-complex symmetric transformation if there exists a conjugation C on Y such that S satisfies the condition S*n∑0≤k≤m(−1)m−kmkS*kCSm−kCSn=0, for some positive integers n and m. This class of transformation contains the class of m-complex symmetric transformations as a proper subset. Some basic structural properties of n-quasi-m-complex symmetric linear transformations are established with the help of transformation matrix representation. In particular, we obtain that a power of an n-quasi-m-complex symmetric is again an n-quasi-m-complex symmetric operator. Moreover, if T and S are such that T is an n1-quasi-m1-complex symmetric and S is an n2-quasi-m2-complex symmetric, their product TS is an max{n1,n2}-quasi-(m1+m2−1)-complex symmetric under suitable conditions. We examine the stability of n-quasi-m-complex symmetric operators under perturbation by nilpotent operators.

Some integrals for groups of bounded linear operators on finite-dimensional non-Archimedean Banach spaces

In this paper, we extend the Volkenborn integral and Shnirelman integral for groups of bounded linear operators on finite-dimensional non-Archimedean Banach spaces over $\mathbb{Q}_{p}$ and $\mathbb{C}_{p}$ respectively. When the ground field is a complete non-Archimedean valued field, which is also algebraically closed, we give some functional calculus for groups of infinitesimal generator $A$ such that $A$ is a nilpotent operator on finite-dimensional non-Archimedean Banach spaces.

On perturbation of Drazin invertible linear relations

UDC 517.98 We study the stability of regular, finite ascent, and finite descent linear relations defined in Banach spaces under commuting nilpotent operator perturbations. As an application, we give the invariance theorem of Drazin invertible spectrum under these perturbations. We also focus on the study of some properties of the left and right Drazin invertible linear relations. It is proved that, for bounded and closed left (resp., right) Drazin invertible linear relation with nonempty resolvent set, 0 is an isolated point of the associated approximate point spectrum (resp., surjective spectrum).

On decomposition of left-right Atkinson linear relations and related classes

ABSTRACT In this paper, left Atkinson linear relations defined in a Banach space are considered. The behaviour of such notion in polynomials is studied and a spectral mapping theorem for the corresponding spectrum is deduced. Let denote any of the following classes: left (right) Atkinson, Fredholm, left (right) Weyl, Weyl, left (right) Browder, Browder linear relations. For a closed and everywhere defined linear relation we prove that under suitable conditions, where and is a bounded nilpotent operator if and only if admits a Kato-decomposition and 0 is not an interior point of the spectrum of As an application, some connections results between and are established, where denotes the Kato-decomposition spectrum of

Exponentially m-isometric operators on Hilbert spaces

For a positive integer m, a bounded linear operator T on a Hilbert space is called an exponentially m-isometric operator if ∑k=0m(−1)m−k(mk)ekT⁎ekT=0. For 1≤n≤m, skew-n-selfadjoint operators, nilpotent operators of order less than or equal to [m+12], the greatest integer not greater than m+12, and 2πi multiples of idempotents are main examples of such operators. We establish a decomposition theorem for strict exponentially m-isometric operators with finite spectrum and prove that they are exponentially isometric m-Jordan. Finally, the dynamics of this operator will be considered. We will show that there is no N-supercyclic exponentially m-isometric operator on an infinite-dimensional Hilbert space.

Commutant hypercyclicity of Hilbert space operators

An operator T on a Hilbert space H is commutant hypercyclic if there is a vector x in H such that the set {Sx : TS = ST} is dense in H. We prove that operators on finite dimensional Hilbert space, a rich class of weighted shift operators, isometries, exponentially isometries and idempotents are all commutant hypercyclic. Then we discuss on commutant hypercyclicity of 2 ? 2 operator matrices. Moreover, for each integer number n ? 2, we give a commutant hypercyclic nilpotent operator of order n on an infinite dimensional Hilbert space. Finally, we study commutant transitivity of operators and give necessary and sufficient conditions for a vector to be a commutant hypercyclic vector.

Non-Abelian Gauge Theories with Composite Fields in the Background Field Method

Non-Abelian gauge theories with composite fields are examined in the background field method. Generating functionals of Green’s functions for a Yang–Mills theory with composite and background fields are introduced, including the generating functional of vertex Green’s functions (effective action). The corresponding Ward identities are obtained, and the issue of gauge dependence is investigated. A gauge variation of the effective action is found in terms of a nilpotent operator depending on the composite and background fields. On-shell independence from the choice of gauge fixing for the effective action is established. In the study of the Ward identities and gauge dependence, finite field-dependent BRST transformations with a background field are introduced and employed on a systematic basis. On the one hand, this involves the consideration of (modified) Ward identities with a field-dependent anticommuting parameter, also depending on a non-trivial background. On the other hand, the issue of gauge dependence is studied with reference to a finite variation of the gauge Fermion. The concept of a joint introduction of composite and background fields to non-Abelian gauge theories is exemplified by the Gribov–Zwanziger theory, including the case of a local BRST-invariant horizon, and also by the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion.

Hammocks to visualize the support of finitely presented functors

Many properties of a module can be expressed in terms of the dimension of the vector space obtained by applying a finitely presented functor to that module. For example, the dimension of the kernel, image or cokernel of the multiplication map given by an algebra element; or the number of summands of a certain type when the module is considered as a module over a subalgebra. When the indecomposable modules over the algebra are arranged in the Auslander-Reiten quiver, the support of the finitely presented functor typically has the shape of a hammock, spanned between sources and sinks. There may also be tangents which are meshes where the hammock function at the middle term exceeds the sum of the values at the start and end terms. We describe how sources, sinks and tangents of the hammock relate to the modules which define the projective resolution of the finitely presented functor. The key tool is the Cokernel Complex Lemma which links the values of the hammock function to the Auslander-Reiten structure of the category. We are also interested in exact subcategories of module categories which have Auslander-Reiten sequences. Our examples include quiver representations and invariant subspaces of nilpotent linear operators.

Which set is the numerical range of an operator?

In this survey paper, we consider the problem of which nonempty bounded convex subset of the complex plane is the numerical range of some bounded linear operator on a complex separable Hilbert space. We start in Section 1 with general operators and move subsequently to operators in certain special classes such as normal and hyponormal operators in Section 2, Toeplitz operators in Section 3, Hankel operators in Section 4, compact operators in Section 5, nilpotent operators and roots of identity in Section 6, Sn-matrices and companion matrices in Section 7, and, finally, nonnegative matrices in Section 8. The known main results will be briefly sketched, which are interspersed with relevant unsolved problems.

Operators and Graph Isomorphism

The article considers a method for recognizing graph isomorphism. The basis of the method is the construction of a chain of transformations of the adjacency matrix of the edge graph L(G) using the properties of the nilpotent operator of the subgraph space. Traditional approaches to graph structure analysis are basically grounded on the use of the adjacency matrix A(G) of the graph. In this article, we propose an approach based on the use of the adjacency matrix of the edge graph A(L(G)) considered as a linear operator of the subgraph space of the graph G. The spectrum of elements of the subgraph space of the graph characterizing the set of edge graph cuts. 7he concept of an edge cut of a graph is introduced and described in the article. It is shown that the formation of a chain of edge cuts generates a spectrum of graph edge cuts, which, in turn, determines the metric properties of the edges and vertices of the graph. The metric properties of the spectrum of edge cuts allow constructing numerical characteristics of the graph structure and the invariant of graph edge cuts, as well as applying it to the solution of the graph isomorphism recognition problem. In this paper, we consider the main algorithmic properties of the spectrum of graph edge cuts and propose an approach for constructing an invariant of edge cuts. An algorithm for determining the invariant of graph edge cuts based on the concept of the spectrum of edge cuts is proposed. It is shown that the construction of an invariant of edge cuts of a graph belongs to the class of polynomial problems.

Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case* *Supported by projects PID2019-105138GB-C21/AEI/10.13039/501100011033 (Spain) and ED431C 2019/10 (Xunta de Galicia, Spain).

On a smooth metric measure spacetime (M, g, e −f dvol g ), we define a weighted Einstein tensor. It is given in terms of the Bakry–Émery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the density function. We consider the associated vacuum weighted Einstein field equations and show that isotropic solutions have nilpotent Ricci operator. Moreover, the underlying manifold is a Brinkmann wave if it is two-step nilpotent and a Kundt spacetime if it is three-step nilpotent. More specific results are obtained in dimension 3, where all isotropic solutions are given in local coordinates as plane waves or Kundt spacetimes.

The existence of Hall polynomials for x2-bounded invariant subspaces of nilpotent linear operators

We prove the existence of Hall polynomials for x 2 -bounded invariant subspaces of nilpotent linear operators.