HessenbergOperator - crowlogic/arb4j GitHub Wiki

A Hessenberg operator is defined in the context of linear operators on Hilbert spaces, and its representation in matrix form is what distinguishes it. Let's delve into the details with formulas:

Formal Definition

Consider a Hilbert space $H$ and a linear operator $T: H \rightarrow H$. $T$ is called a Hessenberg operator if, for some orthonormal basis ${e_n}_{n=1}^{\infty}$ of $H$, the matrix representation of $T$ is a Hessenberg matrix.

Hessenberg Matrix

A Hessenberg matrix is an infinite matrix $[T_{ij}]$ where $T_{ij} = \langle Te_j, e_i \rangle$ and it has the following form:

  1. Upper Hessenberg Matrix: All entries below the first subdiagonal are zero. Formally, for $i > j + 1$, $T_{ij} = 0$.

$$ \begin{pmatrix} T_{11} & T_{12} & T_{13} & \cdots \ T_{21} & T_{22} & T_{23} & \cdots \ 0 & T_{32} & T_{33} & \cdots \ 0 & 0 & T_{43} & \cdots \ \vdots & \vdots & \vdots & \ddots \end{pmatrix} $$

  1. Lower Hessenberg Matrix: All entries above the first superdiagonal are zero. Formally, for $i < j - 1$, $T_{ij} = 0$.

$$ \begin{pmatrix} T_{11} & T_{12} & 0 & 0 & \cdots \ T_{21} & T_{22} & T_{23} & 0 & \cdots \ T_{31} & T_{32} & T_{33} & T_{34} & \cdots \ \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix} $$

Properties and Applications:

  • Spectral Analysis: Spectral properties of $T$ can be studied through its Hessenberg matrix representation.
  • Eigenvalue Problems: Hessenberg matrices are used in numerical methods for finding eigenvalues of large matrices.
  • Numerical Stability: The Hessenberg form is more numerically stable than other forms for certain computations.

Example:

In a simple finite-dimensional case, consider a 3x3 matrix:

$$ \begin{pmatrix} 1 & 4 & 7 \ 2 & 5 & 8 \ 0 & 3 & 6 \end{pmatrix} $$

This is an example of a lower Hessenberg matrix. The concept extends similarly to infinite-dimensional operators.

In summary, Hessenberg operators are defined by their matrix representation in an orthonormal basis, either upper or lower Hessenberg, and have significant applications in numerical analysis and linear algebra.