Spectrum of a non -self-adjoint differential operators with block-triangular matrix coefficients

10.12737/4780 ◽  
2014 ◽  
Vol 2 (4) ◽  
pp. 360-364
Author(s):  
A. Holkin
Keyword(s):  
Differential Operators ◽  
Triangular Matrix ◽  
Matrix Coefficients ◽  
Block Triangular Matrix

SPHERICAL FUNCTIONS, THE COMPLEX HYPERBOLIC PLANE AND THE HYPERGEOMETRIC OPERATOR

2006 ◽  
Vol 17 (10) ◽  
pp. 1151-1173 ◽  
Author(s):  
P. ROMÁN ◽  
J. TIRAO
Keyword(s):  
Hypergeometric Function ◽  
Hyperbolic Plane ◽  
Casimir Operator ◽  
Differential Operators ◽  
Real Variable ◽  
Spherical Functions ◽  
Matrix Coefficients ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Complex Hyperbolic Plane

In this paper, we determine all irreducible spherical functions Φ of any K-type associated to the dual Hermitian symmetric pairs (G, K) = ( SU (3), U (2)) and ( SU (2,1), U (2)). This is accomplished by associating to Φ a vector valued function H = H(u) of a real variable u, analytic at u = 0, which is a simultaneous eigenfunction of two second order differential operators with matrix coefficients. One of them comes from the Casimir operator of G and we prove that it is conjugated to a hypergeometric operator, allowing us to express the function H in terms of a matrix valued hypergeometric function. For the compact pair ( SU (3), U (2)), this project was started in [4].

SIMILARITY OF OPERATORS ON TENSOR PRODUCTS OF SPACES AND MATRIX DIFFERENTIAL OPERATORS

2018 ◽  
Vol 106 (1) ◽  
pp. 19-30
Author(s):  
MICHAEL GIL’
Keyword(s):  
Differential Operators ◽  
Normal Operator ◽  
Variable Coefficients ◽  
Closed Convex Hull ◽  
Constant Matrix ◽  
Norm Estimates ◽  
Matrix Coefficients ◽  
Matrix Differential Operators ◽  
Constant Coefficients ◽  
Matrix Differential

Let ${\mathcal{H}}=\mathbb{C}^{n}\otimes {\mathcal{E}}$ be the tensor product of a Euclidean space $\mathbb{C}^{n}$ and a separable Hilbert space ${\mathcal{E}}$. Our main object is the operator $G=I_{n}\otimes S+A\otimes I_{{\mathcal{E}}}$, where $S$ is a normal operator in ${\mathcal{E}}$, $A$ is an $n\times n$ matrix, and $I_{n},I_{{\mathcal{E}}}$ are the unit operators in $\mathbb{C}^{n}$ and ${\mathcal{E}}$, respectively. Numerous differential operators with constant matrix coefficients are examples of operator $G$. In the present paper we show that $G$ is similar to an operator $M=I_{n}\otimes S+\hat{D}\times I_{{\mathcal{E}}}$ where $\hat{D}$ is a block matrix, each block of which has a unique eigenvalue. We also obtain a bound for the condition number. That bound enables us to establish norm estimates for functions of $G$, nonregular on the closed convex hull $\operatorname{co}(G)$ of the spectrum of $G$. The functions $G^{-\unicode[STIX]{x1D6FC}}\;(\unicode[STIX]{x1D6FC}>0)$ and $(\ln G)^{-1}$ are examples of such functions. In addition, in the appropriate situations we improve the previously published estimates for the resolvent and functions of $G$ regular on $\operatorname{co}(G)$. Since differential operators with variable coefficients often can be considered as perturbations of operators with constant coefficients, the results mentioned above give us estimates for functions and bounds for the spectra of differential operators with variable coefficients.

Graded involutions on block-triangular matrix algebras

2020 ◽  
Vol 585 ◽  
pp. 24-44
Author(s):  
Claudemir Fidelis ◽  
Dimas José Gonçalves ◽  
Diogo Diniz ◽  
Felipe Yukihide Yasumura
Keyword(s):  
Triangular Matrix ◽  
Matrix Algebras ◽  
Block Triangular Matrix

Abelian Gradings on Upper Block Triangular Matrices

2012 ◽  
Vol 55 (1) ◽  
pp. 208-213 ◽  
Author(s):  
Angela Valenti ◽  
Mikhail Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Block Triangular Matrix

AbstractLet G be an arbitrary finite abelian group. We describe all possible G-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

On the factorability of polynomial identities of upper block triangular matrix algebras graded by cyclic groups

2020 ◽  
Vol 601 ◽  
pp. 311-337
Author(s):  
Onofrio Mario Di Vincenzo ◽  
Marcos Antônio da Silva Pinto ◽  
Viviane Ribeiro Tomaz da Silva
Keyword(s):  
Triangular Matrix ◽  
Polynomial Identities ◽  
Matrix Algebras ◽  
Cyclic Groups ◽  
Block Triangular Matrix
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