Generalized inverses of tridiagonal operators

Abstract

Let H be a Hilbert space with $\{e^n:n\in \mathbb{N}\}$ as an orthonormal basis. Let $T:H\to H$ be a bounded linear operator defined by ${\mathit{Te}}^n=e^{n-1}+\lambda \sin (2\mathit{nr})e^n+e^{n+1}$, where λ is real and r is a rational multiple of π. In this short note it is established that the Moore–Penrose inverse of T is not bounded. We also show that the same conclusion is valid for a few related classes of operators.

Introduction

Let H be a separable real Hilbert space. Then H is isometrically isomorphic to ${\ell }^2$, the Hilbert space of square summable real sequences. Let $\{e^n:n\in \mathbb{N}\}$, be the standard orthonormal basis of ${\ell }^2$, where $e^n=(0\text{,}0\text{,}\dots \text{,}1\text{,}0\text{,}\dots )$ with 1 appearing in the nth coordinate. A class of tridiagonal operators defined on ${\ell }^2$ called almost Mathieu operators which are important in Mathematical Physics (see the references cited in [1]) are defined by

$${\mathit{Te}}^n=e^{n-1}+\lambda \sin (2\mathit{nr})e^n+e^{n+1}\text{,}$$

where λ is real and r is a rational multiple of π. The problem of invertibility of such operators was considered recently [1] where it was shown that these operators are not invertible. In this paper, we prove that the situation is worse in the sense that these operators even do not have bounded generalized inverses of two specific types. We also show that the same conclusion holds for some other related classes of infinite matrices.

The existence of generalized inverses of matrices was first observed by Moore who gave the definition of a unique inverse for every finite matrix, square or rectangular. He used the term “general reciprocal”. His publication appeared in 1920, but his results were apparently obtained much earlier. Moore’s discovery was not noticed for about three decades. Revival of interest in the generalized inverse was focussed on the least squares properties, which were recognized by Bjerhammar who rediscovered Moore’s generalized inverse. Penrose in 1955 sharpened and generalized Bjerhammar’s results and showed that Moore’s inverse is the unique solution of the following four equations which have come to be called Penrose equations:

$$\mathit{AXA}=A\text{;}\mathit{XAX}=X\text{;}(\mathit{AX}{)}^{\ast }=\mathit{AX}\text{;}(\mathit{XA}{)}^{\ast }=\mathit{XA}\text{.}$$

We refer the reader to [2], [3] for a more detailed historical introduction.

The problem of determination of the Moore–Penrose inverse of a matrix has been handled by many authors. A classical method of determining the Moore–Penrose inverse was proposed by Decell. This uses the Cayley-Hamilton theorem. Some of the more well known techniques include the formula using the full-rank factorization of a matrix and the method that employs the singular value decomposition. The latter has the advantage of being a numerically more stable procedure. We refer the reader to [3] for more details. A technique which is being widely used in practice was invented by Greville [5]. This is an ingenious iterative method that determines the Moore–Penrose inverse of a matrix by successively adding a column to the given matrix. We refer the reader to [10] for a much simpler constructive proof of the Greville formula and [9] for a proof by verification. Udwadia and Kalaba also provide constructive proofs of other types of generalized inverses [11], [12], [13]. A generalized inverse not so widely studied in the literature is the MP M-inverse of a matrix. This was introduced by Rao and Mitra [8]. Recently, Udwadia and Phohomsiri gave a recursive formula for computing the MP M-inverse. This notion has been shown to have applications in analytical dynamics (See [14] and the references cited therein).

We next briefly review the notion of the Moore–Penrose inverse of operators between Hilbert spaces. Let H₁ and H₂ be Hilbert spaces and $A:H_1\to H_2$ be a bounded linear map. For a linear map $X:H_2\to H_1$ the Penrose equations (as given above) have a unique solution X denoted by A^†. It is known that A^† is bounded iff R(A), the range space of A is closed in H₂. More generally, A has a bounded $\{1\}-\mathit{inverse}$ (any X satisfying $\mathit{AXA}=A$) iff R(A) is closed [4], [6]. There is another well known type of a generalized inverse, defined for square matrices, called the group inverse. Let $H_1=H_2$. Then the group inverse (if it exists) of A is the unique solution of the equations:

$$\mathit{AXA}=A\text{;}\mathit{XAX}=X\text{;}\mathit{AX}=\mathit{XA}\text{.}$$

It is well known that ([3], [7]) for a finite square matrix A, the group inverse denoted by A^# exists iff $R(A)=R(A^2)$ and $N(A)=N(A^2)$. Robert [7] has shown that for a linear map A over a vector space, the group inverse exists iff R(A) and N(A) are complementary subspaces. In particular, if A is one–one, then A^# exists iff A is onto, in which case, $A^{\#}=A^{-1}$. From the discussion as above, it is clear that for a bounded linear operator A, the group inverse A^# is bounded iff R(A) is closed.