Jacobi operator

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 operatorsEdit

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  . In this case it is given by

 

where the coefficients are assumed to satisfy

 

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   of the recurrence relation

 

is a polynomial of degree n and these polynomials are orthonormal with respect to the spectral measure corresponding to the first basis vector  .

This recurrence relation is also commonly written as

 

ApplicationsEdit

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:

GeneralizationsEdit

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

 

and  . 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

 

The zeros of the Bergman polynomial   correspond to the eigenvalues of the principal   submatrix of D. That is, The Bergman polynomials are the characteristic polynomials for the principal submatrices of the shift operator.

ReferencesEdit

  1. ^ Meurant, Gérard; Sommariva, Alvise (2014). "Fast variants of the Golub and Welsch algorithm for symmetric weight functions in Matlab" (PDF). Numerical Algorithms. 67 (3): 491–506. doi:10.1007/s11075-013-9804-x. S2CID 7385259.
  2. ^ Tomeo, V.; Torrano, E. (2011). "Two applications of the subnormality of the Hessenberg matrix related to general orthogonal polynomials". Linear Algebra and Its Applications. 435 (9): 2314–2320. doi:10.1016/j.laa.2011.04.027.
  3. ^ Saff, Edward B.; Stylianopoulos, Nikos (2012). "Asymptotics for Hessenberg matrices for the Bergman shift operator on Jordan regions". arXiv:1205.4183. {{cite journal}}: Cite journal requires |journal= (help)
  4. ^ Escribano, Carmen; Giraldo, Antonio; Asunción Sastre, M.; Torrano, Emilio (2011). "The Hessenberg matrix and the Riemann mapping". arXiv:1107.6036. {{cite journal}}: Cite journal requires |journal= (help)

External linksEdit