Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

Linear Combinations ◽

Special Cases ◽

Sufficient And Necessary Conditions ◽

Alpha Beta ◽

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

Linear combinations of polynomials with three-term recurrence

Publications de l Institut Mathematique ◽

10.2298/pim2124029t ◽

2021 ◽

Vol 110 (124) ◽

pp. 29-40

Author(s):

Khang Tran ◽

Maverick Zhang

Keyword(s):

Linear Combination ◽

Chebyshev Polynomials ◽

Sufficient Conditions ◽

Linear Combinations ◽

Zero Distribution ◽

Necessary And Sufficient ◽

Linear Polynomials

We study the zero distribution of the sum of the first n polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of az + b for a, b ? R, of Chebyshev polynomials. In particular, we find necessary and sufficient conditions on a, b such that this linear combination is hyperbolic.

Necessary and sufficient conditions on weighted multilinear fractional integral inequality

Communications on Pure and Applied Analysis ◽

10.3934/cpaa.2021196 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Zhou Yongliang ◽

Deng Yangkendi ◽

Wu Di ◽

Yan Dunyan

Keyword(s):

Integral Inequality ◽

Sobolev Inequality ◽

Fractional Integral ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Integral Inequalities ◽

Multilinear Fractional Integral ◽

Sufficient And Necessary Conditions ◽

<p style='text-indent:20px;'>We consider certain kinds of weighted multi-linear fractional integral inequalities which can be regarded as extensions of the Hardy-Littlewood-Sobolev inequality. For a particular case, we characterize the sufficient and necessary conditions which ensure that the corresponding inequality holds. For the general case, we give some sufficient conditions and necessary conditions, respectively.</p>

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

Transfer functions and conditions for stationarity of bilinear models with gaussian residuals

10.1098/rspa.1985.0082 ◽

1985 ◽

Vol 400 (1819) ◽

pp. 315-330 ◽

Cited By ~ 10

Keyword(s):

Transfer Function ◽

Transfer Functions ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Function System ◽

Bilinear Models ◽

Special Cases ◽

The aim of this paper is to construct the transfer function system of bilinear models with gaussian residuals and to give necessary conditions and sufficient conditions for stationarity. In some special cases, the necessary and sufficient conditions are given.

NECESSARY AND SUFFICIENT CONDITIONS FOR FREQUENT CAUCHY SEQUENCES

Asian-European Journal of Mathematics ◽

10.1142/s1793557109000248 ◽

2009 ◽

Vol 02 (02) ◽

pp. 295-305

Author(s):

Chuan Jun Tian ◽

Sui Sun Cheng ◽

Mehmet Gürdal

Keyword(s):

Necessary Conditions ◽

Sufficient Conditions ◽

Asymptotic Density ◽

Frequency Measure ◽

Cauchy Sequences ◽

Natural Density ◽

Sufficient And Necessary Conditions ◽

In this paper, by means of frequency measure (also called asymptotic density or natural density) defined for integer subsets, we obtain some new explicit sufficient and necessary conditions for real sequences to be frequently convergent. These conditions involve frequent Cauchy sequences and statistical pre-Cauchy sequences as well as statistical Cauchy sequences.

A Note on Parametric Surfaces in Minkowski 3-Space

The Scientific World JOURNAL ◽

10.1155/2014/618340 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Esra Betul Koc Ozturk

Keyword(s):

Linear Combination ◽

Sufficient Conditions ◽

Ruled Surface ◽

Frenet Frame ◽

Parametric Surface ◽

Parametric Surfaces ◽

Null Curve ◽

With the help of the Frenet frame of a given pseudo null curve, a family of parametric surfaces is expressed as a linear combination of this frame. The necessary and sufficient conditions are examined for that curve to be an isoparametric and asymptotic on the parametric surface. It is shown that there is not any cylindrical and developable ruled surface as a parametric surface. Also, some interesting examples are illustrated about these surfaces.

On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

Mathematica Slovaca ◽

10.1515/ms-2021-0059 ◽

2021 ◽

Vol 71 (6) ◽

pp. 1375-1400

Author(s):

Feyzi Başar ◽

Hadi Roopaei

Keyword(s):

Lower Bound ◽

Sequence Space ◽

Sufficient Conditions ◽

Matrix Transformations ◽

Infinite Matrix ◽

The Matrix ◽

Alpha Beta ◽

Necessary And Sufficient ◽

Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

Normal and osculating maps for submanifolds of RN

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500024252 ◽

1990 ◽

Vol 114 (1-2) ◽

pp. 39-55 ◽

Cited By ~ 3

Author(s):

I. Cattaneo Gasparini ◽

G. Romani

Keyword(s):

Isometric Immersion ◽

Normal Bundle ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Gauss Map ◽

Totally Geodesic ◽

Precise Formulation ◽

Parallel Case ◽

SynopsisLet Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .

AN W ALGEBRAS AND CLASSIFICATION

Modern Physics Letters A ◽

10.1142/s0217732392002822 ◽

1992 ◽

Vol 07 (36) ◽

pp. 3419-3423

Author(s):

LIU CHAO ◽

BOYU HOU

Keyword(s):

Lie Algebra ◽

Regular Element ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Arbitrary Degree ◽

Wznw Model ◽

The necessary and sufficient conditions for the existence of a regular element of arbitrary degree under arbitrary integral gradation of the Lie algebra g is presented. Such elements, while chosen as constraints in WZNW model, give rise to a W-algebra. It is then found that there might be some isomorphic relations between different W-algebras. The necessary conditions for such isomorphisms to appear are also given. Up to the A4 cases these conditions are checked to be sufficient.

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

Embedding theorems and the spectra of certain differential operators

10.1098/rspa.1991.0118 ◽

1991 ◽

Vol 434 (1892) ◽

pp. 643-656 ◽

Cited By ~ 2

Keyword(s):

Differential Operators ◽

Necessary Conditions ◽

Sufficient Conditions ◽

Independent Interest ◽

Embedding Theorems ◽

This paper is concerned with problems of the form n Ʃ k =0 (─1) k ( ρ 2 k y ( k ))( k ) = λ r 2 y on R , ry ∈ L 2 ( R ) and gives conditions on the coefficients sufficient to ensure that the spectrum is discrete; necessary conditions are also given. In certain circumstances, necessary and sufficient conditions for discreteness are given, thus extending the celebrated Molcanov criterion. These results follow from embedding theorems which have independent interest.

Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

Algebra Colloquium ◽

10.1142/s1005386709000297 ◽

2009 ◽

Vol 16 (02) ◽

pp. 293-308 ◽

Cited By ~ 13

Author(s):

Qingwen Wang ◽

Guangjing Song ◽

Xin Liu

Keyword(s):

Division Ring ◽

Sufficient Conditions ◽

Linear Matrix ◽

Matrix Equations ◽

Common Solution ◽

Special Cases ◽

The Common ◽

Necessary And Sufficient ◽

Linear Matrix Equations

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.