scholarly journals Lacunas for hyperbolic differential operators with constant coefficients I

1970 ◽  
Vol 124 (0) ◽  
pp. 109-189 ◽  
Author(s):  
M. F. Atiyah ◽  
R. Bott ◽  
L. Gårding
Keyword(s):  
Differential Operators ◽  
Constant Coefficients

scholarly journals Comparison of iterates of a class of differential operators in Roumieu spaces

2017 ◽  
Vol 101 (115) ◽  
pp. 261-266
Author(s):  
Rachid Chaili ◽  
Tayeb Mahrouz
Keyword(s):  
Differential Operators ◽  
Hypoelliptic Operators ◽  
Constant Coefficients

Considering a class of differential operators with constant coefficients including the hypoelliptic operators, we show that the comparison of the operators implies the inclusion between their spaces of Roumieu vectors.

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.

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