Adjoint Elements Research Articles

ON MULTIPLE COMPLETENESS OF A SYSTEM OF EIGEN AND ADJOINT ELEMENTS OF OPERATOR SHEAF IN HILBERT SPACES

In this paper the conditions of the existence of multiple completeness of a system of eigen and adjoint elements of operator sheaf quadratically depending on the spectral parameter in Hilbert space is given.

Perfect colouring of the graph with its kinds

The Perfect colouring (jp) of a graph is an assignment of colours to all elements (vertices, edges, and regions) of the graph such that no two adjoint elements receive the same colour. In this paper, we determined the tight bounds of perfect colouring as/“(G) </P(G) </“(G) + 4, where/“(G) is total colouring of the graph G. Depending on these bounds, the perfect colouring is divided into five different kinds, and the results of these for some standard graphs are presented in the paper.

Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

On the algebra of parallel endomorphisms of a pseudo-Riemannian metric

On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End(TM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra, A is the sum of its radical and of a semi-simple algebra S. We show in arXiv:1402.6642 that S may be of eight different types, including the generic type S=R.Id, and the K\"ahler and hyperk\"ahler types where S is respectively isomorphic to the complex field C or to the quaternions H. We show here that for any self adjoint nilpotent element N of the commutant of such an S in End(TM), the set of germs of metrics such that A contains S and {N} is non-empty. We parametrise it. Generically, the holonomy algebra of those metrics is the full commutant of $S\cup\{N\}$ in O(g). Apart from some "degenerate" cases, the algebra A is then $S \oplus (N)$, where (N) is the ideal spanned by N. To prove it, we introduce an analogy with complex Differential Calculus, the ring R[X]/(X^n) replacing the field C. This describes totally the local situation when the radical of A is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal.

Adjoint of a Linear Multiport Element Based on Generalized Duality

In this brief, a direct and universal method of finding the adjoint of a multiport element is proposed. This is achieved by expressing the characteristic equations of the element as a hybrid-transmission matrix, which is based on the concept of generalized duality, and deriving a simple relation between the hybrid-transmission matrices of the original and adjoint element. The proposed method unifies the existing transpose relations based on conventional admittance, impedance, and hybrid parameter matrices and can be further generalized to cases which existing methods cannot conveniently deal, or even deal, with. Once the characteristic equations of the element are given, we can obtain the adjoint (if it does exist) immediately by using the proposed method without any additional operations. The proofs of relevant theorems and some applications of the proposed method are given and the sensitivity formula based on the proposed method is also derived.

Approximately Quasi Inner Generalized Dynamics on Modules

We investigate some properties of approximately quasi inner generalized dynamics and quasi approximately inner generalized derivations on modules. In particular, we prove that if A is a C*-algebra, is the generator of a generalized dynamics on an A-bimodule M satisfying and there exist two sequences of self adjoint elements in A such that for all in a core for , , then is approximately quasi inner.

Problem of small and normal oscillations of a rotating elastic body filled with an ideal barotropic liquid

An evolutionary problem of small motions of an ideal barotropic liquid filling a rotating isotropic elastic body is studied in the paper. Moreover, the corresponding spectral problem arising in the study of normal motions of the mentioned system is considered. First, we state the evolutionary problem, then we pass to a second-ordered differential equation in some Hilbert space. Based on this equation, we prove the uniqueness theorem for the strong solvability of the corresponding mixed problem. The spectral problem is studied in the second part of the paper. A quadratic spectral sheaf corresponding to the spectral problem was derived and studied. Problems of localization, discreteness, and asymptotic form of the spectrum are considered for this sheaf. The statement of double completeness with a defect for a system of eigenelements and adjoint elements and the statement of essential spectrum of the problem are proved.

On a reduction procedure for Horn inequalities in finite von Neumann algebras

We consider the analogues of the Horn inequalities in finite von Neumann algebras, which concern the possible spectral distributions of sums $a+b$ of self--adjoint elements $a$ and $b$ in a finite von Neumann algebra. It is an open question whether all of these Horn inequalities must hold in all finite von Neumann algebras, and this is related to Connes' embedding problem. For each choice of integers $1\le r\le n$, there is a set $T^n_r$ of Horn triples, and the Horn inequalities are in one-to-one correspondence with $\cup_{1\le r\le n}T^n_r$. We consider a property P$_n$, analogous to one introduced by Therianos and Thompson in the case of matrices, amounting to the existence of projections having certain properties relative to arbitrary flags, which guarantees that a given Horn inequality holds in all finite von Neumann algebras. It is an open question whether all Horn triples in $T^n_r$ have property P$_n$. Certain triples in $T^n_r$ can be reduced to triples in $T^{n-1}_r$ by an operation we call {\em TT--reduction}. We show that property P$_n$ holds for the original triple if property P$_{n-1}$ holds for the reduced one. We then characterize the TT--irreducible Horn triples in $T^n_3$, for arbitrary $n$, and for those LR--minimal ones (namely, those having Littlewood--Richardson coefficient equal to 1), we perform a construction of projections with respect to flags in arbitrary von Neumann algebras in order to prove property P$_n$ for them. This shows that all LR--minimal triples in $\cup_{n\ge3}T^n_3$ have property P$_n$, and so that the corresponding Horn inequalities hold in all finite von Neumann algebras.

On Left–Right Consistency in Rings

In a C*-algebra the spectrum of the product of a positive and a self adjoint element is always real. This simple observation is the tip of a curious iceberg, built on a sort of 'left-right consistency' relative to invert ibility. Suppose A is a semigroup, assumed by default to have an identity 1, with invertible group A~l Aj^tnA~^ght' more generally much of what we have to say extends to an abstract category. Elements a e A induce left and right multiplications on ^4, La : x i> ax ; Ra : x h-» xa . It is the relationship between these operators that gives rise to the left-right consistency behind the positive operator phenomenon.

Current operational amplifiers: Basic architecture, properties, exploitation and future

This review paper describes all major considerations of the design of current operational amplifiers (COAs). It is demonstrated, that the single-input, differential-output COA as the adjoint element to the differentialinput, single-output voltage operational amplifier (VOA) is employable in current processing circuits alike the VOAs in voltage processing circuits. On an example of a basic one-high-gain-stage current-operational-amplifier architecture the procedure and the strategy of the design is explained and the main first and second order properties of this device are described. The paper is concluded showing elementary approaches of the derivation of the exploitation of the COAs in current processing circuits interreciprocal to voltage processing circuits and discussing the considerations of the exploitation of the COAs in VLSI designs.

On characteristic and adjoint elements of linear operators

The aim of the present paper is to show that with certain assumptions a system of characteristic and adjoint elements of a linear operator can be so chosen that the Gram matrix of this system is finite and has a finite reciprocal matrix. A system of elements with such a Gram matrix will be called normal. Problems of expansion in such systems are solved quite simply and many propositions in the theory of linear equations with non-selfconjugate operators are formulated in almost the same way as in the theory of equations with self-conjugate operators.

Order and commutativity in Banach algebras

S. Sherman has shown [4] that if the self adjoint elements of a C* algebra form a lattice under their natural ordering the algebra is necessarily commutative. In this note we extend this result to real Banach algebras with an identity and arbitrary Banach * algebras with an identity. The central fact for a real Banach algebra A is that if the positive cone is defined to be the uniform closure of the set of finite sums of squares of elements of A, and if A is a lattice under the ordering induced by this cone, then extreme points of the unit sphere of the dual cone are multiplicative linear functionals. A similar situation holds for * algebras.

Structure theorems for a special class of Banach algebras

Introduction. In this paper we prove analogues of two of the Wedderburn structure theorems for a special class of algebras (we prefer the term Banach to the more common term normed ring). Our theorems assert, first, that any algebra of the special kind we shall consider is a direct sum of simple ones (simple no closed 2-sided ideals) and, secondly, that every simple algebra of this type is a full matrix algebra-where by a full matrix algebra we mean the set of all finite or infinite (countable or uncountable) matrices for which the sum of the squares of the absolute values of the elements converges (the matrix elements being complex numbers). We use the term algebra, or simply B-algebra, for a set which is an algebra over the complex numbers (though without any assumptions about a basis, finite or otherwise) whose underlying linear space has a norm with respect to which it is a space, and which satisfies the condition ||xy|| < X IyI . These assumptions are the same as those of Gelfand [III] (I ) except that we assume neither commutativity nor the existence of a unit. The special algebras which we consider (we call them H*-algebras) satisfy the additional conditions: (1) the underlying space is a Hilbert space (of arbitrary dimension), (2) each element x has an x* in a certain rather strong sense. The part played in our theory by the fact that the underlying space is a Hilbert space is two-fold. It makes direct sum decompositions easier through the possibility of taking orthogonal complements, and it opens up the possibility of using the spectral theory of operators on Hilbert space. Actually we do not use the spectral theory but instead use simplified forms of some of the technique used in that theory. It is here that the adjoint elements in our algebra are essential for their existence makes it possible, through the spectral theory technique, to construct idempotents. Our consideration of these H*-algebras arose from a consideration of the L2-algebra of a compact group. Segal [VIII] has defined the group algebra of a locally compact group to be the space L1 of integrable functions (that is, complex-valued functions, integrable with respect to the Haar measure of the group) with convolution for multiplication. In the case of a compact group the space L2 (of complex-valued functions whose square is integrable with