A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel measure. This operator is named after Carl Gustav Jacob Jacobi.

The name derives from a theorem from Jacobi, dating to 1848, stating that every symmetric matrix over a principal ideal domain is congruent to a tridiagonal matrix.

Self-adjoint Jacobi operators

The most important case is the one of self-adjoint Jacobi operators acting on the Hilbert space of square summable sequences over the positive integers ${\displaystyle \ell ^{2}(\mathbb {N} )}$. In this case it is given by

${\displaystyle Jf_{0}=a_{0}f_{1}+b_{0}f_{0},\quad Jf_{n}=a_{n}f_{n+1}+b_{n}f_{n}+a_{n-1}f_{n-1},\quad n>0,}$

where the coefficients are assumed to satisfy

${\displaystyle a_{n}>0,\quad b_{n}\in \mathbb {R} .}$

The operator will be bounded if and only if the coefficients are bounded.

There are close connections with the theory of orthogonal polynomials. In fact, the solution ${\displaystyle p_{n}(x)}$ of the recurrence relation

${\displaystyle J\,p_{n}(x)=x\,p_{n}(x),\qquad p_{0}(x)=1{\text{ and ))p_{-1}(x)=0,}$

is a polynomial of degree n and these polynomials are orthonormal with respect to the spectral measure corresponding to the first basis vector ${\displaystyle \delta _{1,n))$.

This recurrence relation is also commonly written as

${\displaystyle xp_{n}(x)=a_{n+1}p_{n+1}(x)+b_{n}p_{n}(x)+a_{n}p_{n-1}(x)}$

Applications

It arises in many areas of mathematics and physics. The case a(n) = 1 is known as the discrete one-dimensional Schrödinger operator. It also arises in:

• The Lax pair of the Toda lattice.
• The three-term recurrence relationship of orthogonal polynomials, orthogonal over a positive and finite Borel measure.
• Algorithms devised to calculate Gaussian quadrature rules, derived from systems of orthogonal polynomials.^[1]

Generalizations

When one considers Bergman space, namely the space of square-integrable holomorphic functions over some domain, then, under general circumstances, one can give that space a basis of orthogonal polynomials, the Bergman polynomials. In this case, the analog of the tridiagonal Jacobi operator is a Hessenberg operator – an infinite-dimensional Hessenberg matrix. The system of orthogonal polynomials is given by

${\displaystyle zp_{n}(z)=\sum _{k=0}^{n+1}D_{kn}p_{k}(z)}$

and ${\displaystyle p_{0}(z)=1}$. Here, D is the Hessenberg operator that generalizes the tridiagonal Jacobi operator J for this situation.^[2][3][4] Note that D is the right-shift operator on the Bergman space: that is, it is given by

${\displaystyle [Df](z)=zf(z)}$

The zeros of the Bergman polynomial ${\displaystyle p_{n}(z)}$ correspond to the eigenvalues of the principal ${\displaystyle n\times n}$ submatrix of D. That is, The Bergman polynomials are the characteristic polynomials for the principal submatrices of the shift operator.