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I was wondering how I can define a pseudodifferential operator using Cauchy integral formula.

Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as:

$$\begin{equation} p(T) = \oint_{C} p(z)(zI-T)^{-1}dz \end{equation}$$

where contour $C$ encloses all eigenvalues of $T$ and $I$ is the identity operator.

My question: can a similar definition be used to define $p(x,T)$?

My understanding is that $p(x,T) = \oint_{C} p(x,z)(zI-T)^{-1}dz$ cannot be true in general if $T$ and $x$ do not commute.

Update:

I believe a clarification to this question is necessary given the discussion below. The objective of this question is to find a generalization/modification to Cauchy integral formula to make it applicable to pseduodifferential operators. I am convinced that $p(x,T) = \oint_{C} p(x,z)(zI-T)^{-1}dz$ is not valid in general.

I was wondering how I can define a pseudodifferential operator using Cauchy integral formula.

Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as:

$$\begin{equation} p(T) = \oint_{C} p(z)(zI-T)^{-1}dz \end{equation}$$

where contour $C$ encloses all eigenvalues of $T$ and $I$ is the identity operator.

My question: can a similar definition be used to define $p(x,T)$?

My understanding is that $p(x,T) = \oint_{C} p(x,z)(zI-T)^{-1}dz$ cannot be true in general if $T$ and $x$ do not commute.

Update:

I believe a clarification to this question is necessary given the discussion below. The objective of this question is to find a generalization/modification of Cauchy integral formula to make it applicable to pseduodifferential operators. I am convinced that $p(x,T) = \oint_{C} p(x,z)(zI-T)^{-1}dz$ is not valid in general.

I was wondering how I can define a pseudodifferential operator using Cauchy integral formula.

Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as:

$$\begin{equation} p(T) = \oint_{C} p(z)(zI-T)^{-1}dz \end{equation}$$

where contour $C$ encloses all eigenvalues of $T$ and $I$ is the identity operator.

My question: can a similar definition be used to define $p(x,T)$?

My understanding is that $p(x,T) = \oint_{C} p(x,z)(zI-T)^{-1}dz$ cannot be true in general if $T$ and $x$ do not commute (?).

I was wondering how I can define a pseudodifferential operator using Cauchy integral formula.

Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as:

$$\begin{equation} p(T) = \oint_{C} p(z)(zI-T)^{-1}dz \end{equation}$$

where contour $C$ encloses all eigenvalues of $T$ and $I$ is the identity operator.

My question: can a similar definition be used to define $p(x,T)$?

My understanding is that $p(x,T) = \oint_{C} p(x,z)(zI-T)^{-1}dz$ cannot be true in general if $T$ and $x$ do not commute.

Update:

I believe a clarification to this question is necessary given the discussion below. The objective of this question is to find a generalization/modification to Cauchy integral formula to make it applicable to pseduodifferential operators. I am convinced that $p(x,T) = \oint_{C} p(x,z)(zI-T)^{-1}dz$ is not valid in general.

I was wondering how I can define a pseudodifferential operator using Cauchy integral formula.

Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as:

$$\begin{equation} p(T) = \oint_{C} p(z)(zI-T)^{-1}dz \end{equation}$$

where contour $C$ encloses all eigenvalues of $T$ and $I$ is the identity operator.

My question is can a similar definition be used to define $p(x,T)$?

My understanding is that $p(x,T) = \oint_{C} p(x,z)(zI-T)^{-1}dz$ cannot be true in general if $T$ and $x^n$ do not commute (?).

I was wondering how I can define a pseudodifferential operator using Cauchy integral formula.

Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as:

$$\begin{equation} p(T) = \oint_{C} p(z)(zI-T)^{-1}dz \end{equation}$$

where contour $C$ encloses all eigenvalues of $T$ and $I$ is the identity operator.

My question: can a similar definition be used to define $p(x,T)$?

My understanding is that $p(x,T) = \oint_{C} p(x,z)(zI-T)^{-1}dz$ cannot be true in general if $T$ and $x$ do not commute (?).