Skip to main content
added 177 characters in body
Source Link
ABIM
  • 5.4k
  • 3
  • 19
  • 41

Fix $N>1$. Let $f\in C(\mathbb{R},\mathbb{R})$$f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C(\mathbb{R},\mathbb{R}) &\rightarrow C(\mathbb{R},\mathbb{R}) \\ g & \mapsto g \circ f, \end{aligned} $$$$ \begin{aligned} C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\mathbb{R}) \\ g & \mapsto g \circ f, \end{aligned} $$ is a bounded operator. Here the topology on $C^{\infty}(\mathbb{R},\mathbb{R})$ is the Whitney topology.

Let $C^f$ denote the adjoint operator $C_f$. Is there a criterion on $f$ to verify when $C^f$ has at most $N$ linearly independent eigenvectors?

Fix $N>1$. Let $f\in C(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C(\mathbb{R},\mathbb{R}) &\rightarrow C(\mathbb{R},\mathbb{R}) \\ g & \mapsto g \circ f, \end{aligned} $$ is a bounded operator.

Let $C^f$ denote the adjoint operator $C_f$. Is there a criterion on $f$ to verify when $C^f$ has at most $N$ linearly independent eigenvectors?

Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\mathbb{R}) \\ g & \mapsto g \circ f, \end{aligned} $$ is a bounded operator. Here the topology on $C^{\infty}(\mathbb{R},\mathbb{R})$ is the Whitney topology.

Let $C^f$ denote the adjoint operator $C_f$. Is there a criterion on $f$ to verify when $C^f$ has at most $N$ linearly independent eigenvectors?

removed capitals from title
Source Link
YCor
  • 63.9k
  • 5
  • 187
  • 286

Bound on Numbernumber of Linearly Independent Eigenvectorslinearly independent eigenvectors of Adjointadjoint of Composition Operatorcomposition operator

Fix $N>1$. Let $f\in C(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C(\mathbb{R},\mathbb{R}) &\rightarrow C(\mathbb{R},\mathbb{R}) \\ g & \mapsto g \circ f, \end{aligned} $$ is a bounded operator.

Let $C^f$ denote the adjoint operator $C_f$. Is there a criteriacriterion on $f$ to verify when $C^f$ has at-most most $N$ linearly independent eigenvectors?

Bound on Number of Linearly Independent Eigenvectors of Adjoint of Composition Operator

Fix $N>1$. Let $f\in C(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C(\mathbb{R},\mathbb{R}) &\rightarrow C(\mathbb{R},\mathbb{R}) \\ g & \mapsto g \circ f, \end{aligned} $$ is a bounded operator.

Let $C^f$ denote the adjoint operator $C_f$. Is there a criteria on $f$ to verify when $C^f$ has at-most $N$ linearly independent eigenvectors?

Bound on number of linearly independent eigenvectors of adjoint of composition operator

Fix $N>1$. Let $f\in C(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C(\mathbb{R},\mathbb{R}) &\rightarrow C(\mathbb{R},\mathbb{R}) \\ g & \mapsto g \circ f, \end{aligned} $$ is a bounded operator.

Let $C^f$ denote the adjoint operator $C_f$. Is there a criterion on $f$ to verify when $C^f$ has at most $N$ linearly independent eigenvectors?

Source Link
ABIM
  • 5.4k
  • 3
  • 19
  • 41

Bound on Number of Linearly Independent Eigenvectors of Adjoint of Composition Operator

Fix $N>1$. Let $f\in C(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C(\mathbb{R},\mathbb{R}) &\rightarrow C(\mathbb{R},\mathbb{R}) \\ g & \mapsto g \circ f, \end{aligned} $$ is a bounded operator.

Let $C^f$ denote the adjoint operator $C_f$. Is there a criteria on $f$ to verify when $C^f$ has at-most $N$ linearly independent eigenvectors?