# diagonalizable operator

The expression ”diagonalizable operator” has several meanings in operator theory. The purpose of this entry is to present some commonly used concepts where this terminology appears.

## 0.1 Definition 1

Remark - This is a common definition in linear algebra.

## 0.2 Definition 2

Let $H$ be a finite dimensional Hilbert space. A linear operator $T:H\longrightarrow H$ is said to be diagonalizable if there is an orthonormal basis  of $H$ in which $T$ is represented by a diagonal matrix  .

The above definition is equivalent to: There exists an orthonormal basis of $H$ consisting of eigenvectors of $T$.

Another equivalent definition is: There exists an orthonormal basis $\{e_{1},\dots,e_{n}\}$ of $H$ and values $\lambda_{1},\dots,\lambda_{2}\in\mathbb{C}$ such that

 $T(x)=\sum_{i=1}^{n}\lambda_{i}\langle x,e_{i}\rangle e_{i}$

Remarks -

## 0.3 Definition 3

Let $H$ be a Hilbert space. A bounded linear operator $T:H\longrightarrow H$ is said to be diagonalizable if there exists an orthonormal basis consisting of eigenvectors of $T$.

An equivalent definition is: There exists an orthonormal basis $\{e_{i}\}_{i\in J}$ of $H$ and values $\{\lambda_{i}\}_{i\in J}$ such that

 $T(x)=\sum_{i\in J}\lambda_{i}\langle x,e_{i}\rangle e_{i}$

Remarks -

## 0.4 Definition 4

Let $H$ be a Hilbert space. A linear operator $T:H\longrightarrow H$ is said to be diagonalizable if it is to a multiplication operator (http://planetmath.org/MultiplicationOperatorOnMathbbL22) in some $L^{2}$-space (http://planetmath.org/L2SpacesAreHilbertSpaces), i.e. if there exists

 $T=UM_{f}U^{*}$

where $M_{f}:L^{2}(X)\longrightarrow L^{2}(X)$ is the operator of multiplication (http://planetmath.org/MultiplicationOperator) by $f$

 $M_{f}(\psi)=f.\psi\;\;.$

Remarks -

• If $H=\mathbb{C}^{n}$ the above definition is equivalent to say that $T$ is unitarily diagonalizable (Definition 2). Indeed, we can think of $\mathbb{C}^{n}$ as $L^{2}(\{1,\dots,n\})$ with the counting measure. In this case, multiplication operators correspond to diagonal matrices.

• Diagonalizable operators (in this sense) are necessarily normal operators (since multiplication operators are so). The discussion about the converse result is the content of general versions of the spectral theorem.

Title diagonalizable operator DiagonalizableOperator 2013-03-22 17:33:47 2013-03-22 17:33:47 asteroid (17536) asteroid (17536) 6 asteroid (17536) Definition msc 46C05 msc 47A05 SpectralTheoremForHermitianMatrices unitarily diagonalizable