A note on generalized invertibility and invariants of infinite matrices

The classes of band-dominated operators and the subclass of operators in the Wiener algebra W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {W}}$$\end{document} are known to be inverse closed. This paper studies and extends partially known results of that type for one-sided and generalized invertibility. Furthermore, for the operators in the Wiener algebra W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {W}}$$\end{document} invertibility, the Fredholm property and the Fredholm index are known to be independent of the underlying space lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^p$$\end{document}, 1≤p≤∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p\le \infty $$\end{document}. Here this is completed by the observation that even the kernel and a suitable direct complement of the range as well as generalized inverses of operators in W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {W}}$$\end{document} are invariant w.r.t. p.


Introduction
Consider the spaces l p = l p (Z) of two-sided infinite sequences x = (x i ) i∈Z of complex numbers equipped with the respective p-norms where l 0 denotes the subspace of l ∞ of all sequences (x i ) of entries tending to zero as i → ±∞. Roughly speaking, bounded linear operators A ∈ L(l p ) on these l p can be regarded as infinite matrices, and in view of this interpretation the definition of band-dominated operators is straightforward and easily understood: As a start, notice that every sequence a = (a i ) ∈ l ∞ defines an operator of multiplication Theorem 1.1. Let A ∈ A l p . Then A is invertible from the left (resp. right) if and only if its lower norm ν(A) = inf{ Ax : x = 1} (resp. ν(A * )) is positive. In this case there exists a left (right) inverse B in A l p which is also a generalized inverse for A. The latter means that ABA = A and BAB = B, such that I − AB is a projection parallel to the range im A, i.e. onto a complement of im A, and I − BA is a projection onto the kernel ker A.
Note again that B is a Fredholm regularizer and the two projections are compact (cf. [11]). For an introduction to generalized invertibility see e.g. [4,Chapter 4]. The Wiener algebra Although BO is one and the same operator algebra independent of the underlying space l p , its closure, the algebra A l p of banddominated operators, depends on the choice of p. An important and fruitful superset of BO which on the one hand is still p-invariant but on the other hand also beautifully self-contained in many regards is the so called Wiener algebra. It is defined as the closure of BO w.r.t. the norm · W given by Vol. 96 (2022) A note on generalized invertibility 641 Equipped with · W , W is a Banach algebra and it is well known that: 1. W is a subalgebra of A l p for every p with A L(l p ) ≤ A W , i.e. Wiener operators and their adjoints act as band-dominated operators on all l p (cf. [9, Section 2.5]). 2. If A ∈ W is invertible on one l p -space then it is invertible on every l pspace and the inverse A −1 belongs to W, i.e. W is inverse closed (cf. [9, Theorem 2.5.2]). 3. If A ∈ W is Fredholm on one l p -space then A is Fredholm on every l pspace. In this case the Fredholm index is the same on all these spaces. Moreover there exists a B ∈ W which is a Fredholm regularizer for A on all l p (see [10, Theorem 3 and Corollary 25], as well as [1,2,6,7] for a presentation of the history of these results). Thus one can talk about A ∈ W, its invertibility, its Fredholm property, its inverse A −1 and its adjoint A * independently of the underlying space l p , p ∈ {0} ∪ [1, ∞]. This can be enriched by the following: If A is left (resp. right) invertible on one l p then it is left (right) invertible on all l p . In this case, there exists a B ∈ W which is a one-sided inverse and also a generalized inverse for A on all spaces l p . Theorem 1.2 appeared in large parts already in [3] and somehow triggered the present work. Actually, one can push this further and show the following in Sect. 3: (a) ker A is the same on all l p and there is a subspace K which serves as a complement of im A in all l p , resp. In particular, ker A and K are subspaces of l 1 . (b) There exists a generalized inverse B ∈ W, hence I − BA ∈ W is a projection onto ker A and I − AB ∈ W is a projection parallel to im A.
Thus, besides 1. -3. above, also one-sided invertibility, the kernel and the cokernel of A ∈ W as well as suitable generalized inverses are independent of the underlying space. Generalized sequence spaces, the Hilbert space case and the Moore-Penrose inverse In fact, the above mentioned properties 1. -3. of W have been proved in the cited literature for the much more general setting of Wiener operators on the spaces l p (Z N , X) of X-valued generalized sequences (x i ) i∈Z N ⊂ X with X being a Banach space, not necessarily finite dimensional. Also Theorems 1.1, 1.2 and 1.3 extend to this setting and remain true for Fredholm operators A ∈ A l p , resp. A ∈ W. This is the goal of Sect. 4. In the Hilbert space case a very particular generalized inverse, the Moore-Penrose inverse is available, offers a deeper understanding and will be discussed in this last section as well. Finally, a criterion for one-sided invertibility based on finite discretizations is given.  Proof. If B is a right inverse to A then B * is a left inverse to A * . The latter implies that ν(A * ) > 0. Now, let ν(A * ) > 0. Then [11, Corollary 2.3 and Theorem 4.3] apply to A and show again that A is Fredholm. [10, Theorem 21] yields a generalized inverse B ∈ A l p . Since (I − AB)A = 0 and A is surjective, we find that I − AB = 0, i.e. B is a right inverse. The rest easily follows as above.

One-sided invertible band-dominated operators
This finishes the proof of Theorem 1.1.

Remark 1.
For all p ∈ {0} ∪ (1, ∞), A l p includes the ideal of all compact operators (cf. [7,9,10]). This implies that if there is one regularizer B ∈ A l p then all regularizers are band-dominated in these cases. To check this let C ∈ L(l p ) s.t. I − CA is compact.

Operators in the Wiener class
We continue with the highly interesting operators A in the Wiener class W which, as already mentioned in the introduction, have a couple of properties independent of the underlying space. However, in some situations and for some parts of the proofs it is useful to indicate on which l p -space A ∈ W is considered. This is done by the notation A = A p in what follows.

One-sided invertibility
Lemma 3.1. If A,Ã ∈ W coincide on one l p then they coincide on all l p .
Proof. For n ∈ N let P n denote the canonical projection which truncates (x i ) ∈ l p by the rule We continue with the proof of Theorem 1.2 which states that W is one-sided inverse closed: Proof. Let A = A p ∈ W be left invertible on l p and let D ∈ A l p be a left inverse given by Theorem 1.1. Further, choose a band operator C with D − C A < 1/2. Then This Wiener operator B and the equation BA = I, hence the left invertibility of A, translate to all l p -spaces by the previous lemma. The case of right invertible A is analogous.
Notice that in general not all one-sided inverses of A ∈ W belong to W: For example, let P = χ Z+ I be the projection which maps As already mentioned in the introduction, the study and the results of the present work are related to the results on A l p and W obtained in [3]. More precisely, the following was already shown there by different methods: • The characterization of one-sided invertibility of A ∈ A l p , 1 < p < ∞ in terms of its lower norm [3,Theorem 3.9].

On kernels and cokernels
Let's continue with the study of the kernels of Fredholm operators A ∈ W.
Proof. Recall that A = A 1 is Fredholm on l 1 and set K:= ker A 1 and k:= dim K.
Since A = A p is right invertible and Fredholm on all l p and the index of A p is the same on all l p , we find that dim ker A p = k on all l p . So, since K ⊂ l 1 ⊂ l p for all p, the assertion follows. Thus, we get the following picture for one-sided invertible A ∈ W: Clearly the next question is, whether this extends and holds for Fredholm operators A ∈ W in general. Here comes the proof of the affirmative answer which was already stated in Theorem 1.3a): Proof. By [10, Lemma 24] we can choose a Fredholm operator S k ∈ W with ind S k = k = − ind(A). Then AS k ∈ W is Fredholm of index 0 and, by [10,Corollary 12], there is a decomposition AS k = W +S with an invertible W and a compact S of finite rank which fulfills S −P n SP n → 0 as n → ∞. Since the set of invertible operators is open we find for sufficiently large n thatW :=W + Vol. 96 (2022) A note on generalized invertibility 645 (S − P n SP n ) is still invertible. Since P n SP n in the decomposition AS k = W +P n SP n belongs to W, alsoW ∈ W henceW −1 ∈ W. This gives AB = I+T with B:=S kW −1 ∈ W and a finite rank operator T :=P n SP nW −1 ∈ W. By completely analogous arguments starting with S k A one getsBA = I +T , witĥ B ∈ W and a finite rank operatorT ∈ W withT =T P n (without loss of generality with the same n). Next, defineṼ = V 2n+1 and furtherP : ( Then UU = I, U U = I − P n , and im T ⊂ im P n = ker U as well as im U = ker P n ⊂ kerT . We get UABU = I, thus UA ∈ W is Fredholm and right invertible, hence ker(UA) ⊂ l 1 is independent of p by Proposition 3.2. We particularly conclude that ker A ∞ ⊂ ker(UA ∞ ) ⊂ l 1 . Therefore this ker A ∞ is included in ker A p for every p. Since the converse inclusion is obvious, we find that ker A p is the same for every p.
Further UBAU = I, hence AU ∈ W is Fredholm and left invertible andK ⊂ l 1 shall denote the common p-invariant complement of the ranges im AU given by Proposition 3.3. Now consider im A 1 which is a superset of im A 1 U , set K 1 := im A 1 ∩K and fix a decompositionK = K 1 ⊕ K.
Since neither the index of A p nor its kernel dimension depend on p, by the above, the codimension of im A p is p-invariant as well, so that these spaces K p actually coincide with K 1 . Thus K serves as a complement for all im A p .

On generalized inverses
Having proved the p-invariance of the kernel and a complement of the range im A of A ∈ W, we finally turn to some corresponding p-invariant projections in W onto these spaces and a generalized inverse of A in W. We start with an auxiliary result: Thus we have a representation/decomposition of R as a finite sum, where the functionals g k can be interpreted as dual products with certain sequences (g (k) i ) i∈Z ∈ l 1 . To check that R ∈ W it suffices to show that the summands of the form G : x → g(x)y, with g = (g i ) and y = (y i ) of l 1 -type, belong to W: The entries of the canonical matrix representation of G are bounded by |y i ||g j |, hence the elements on the kth diagonal are bounded by (|y i ||g i+k |) i , resp. Consequently, the Wiener norm G W can be estimated as follows: which yields the assertion.
Now we can close this section with the proof of Theorem 1.3b): Proof. Consider A = A 0 on l 0 . In view of Theorem 1.3a) we have ker A 0 ⊂ l 1 and a complement K ⊂ l 1 of im A 0 in l 0 . Choose a projection P 1 from l 0 onto K parallel to im A 0 and a projection P 2 onto ker A 0 . 2 As in the proof of Lemma 3.5 there are representations for P 1 , P 2 ∈ W f j (x)y j and P 2 x = l j=1 g j (x)z j with k = dim coker A, l = dim ker A and f j , g j , y j , z j of l 1 -type. Then the operator R Rx:= min{k,l} j=1 g j (x)y j is compact, belongs to W by Lemma 3.5 and has full rank min{k, l}.
Since g j • P 2 = g j and P 1 y j = y j we have R = P 1 R = RP 2 = P 1 RP 2 . By construction, A + R is Fredholm with dim ker(A + R) = 0 if k ≥ l (resp. dim coker(A + R) = 0 if k ≤ l), thus A + R is one-sided invertible. Let D be a generalized inverse (and hence one-sided inverse) in W for A + R which exists by Theorem 1.

l p (Z N , X) with a Banach space X
We now turn our attention to the generalizations of the above concepts and results to the spaces l p (Z N , X) of Banach space valued generalized sequences (x i ) i∈Z N ⊂ X, where N ∈ N and X is a Banach space. The p-norms are naturally extended as Clearly in this setting the operators of multiplication are of the form aI with bounded a = (a i ) i∈Z N , where a i ∈ L(X), the shifts are V α : (x i ) → (x i−α ) for every α ∈ Z N , and then the definitions of A l p and W are identical. The definition of P n is naturally extended by P n = χ [−n,n] N I. As long as N = 1 and X is of finite dimension, nothing changes, and reusing the above arguments verbatim still gives Theorems 1.1, 1.2 and 1.3. Essentially, the argument which gets lost in the more general situation is the automatic Fredholm property of semi-Fredholm band-dominated operators from [11]. However, supposing additionally that A is Fredholm, the proofs in Sects. 2 and 3.1 still work and immediately yield: For the extension of Theorem 1.3 for Fredholm A ∈ W one starts as in Sect. 3.2 in order to find the relations AB = I + T andBA = I +T with B,B ∈ W and finite rank operators T,T ∈ W such that im T ⊂ im P n and ker P n ⊂ kerT .
Only the definitions of U and U require a modification. For this introduce the operators C j : l p (Z N , X) → X, (x i ) → x j , as well as D j : X → l p (Z N , X) which map x ∈ X to (x i ) with x j = x and x i = 0 for all i = j. Further define the set H:={−n, . . . , n} N ⊂ Z N and two subspaces of X by Note that X 1 is of finite dimension and X 2 of finite codimension in X. Choose a decomposition X = Y ⊕ X 2 and further X = Y ⊕ ((X 1 ∩ X 2 ) ⊕ Z) with Z being of finite codimension. Thus there is a bounded finite rank projection R of X parallel to Z onto Y ⊕ (X 1 ∩ X 2 ). Next, set α:=(2n + 1, 0, . . . , 0) ∈ Z N , define the shiftṼ = V α and the projectionP :

l p (Z N , X) with a Hilbert space X
If X is a Hilbert space then l 2 (Z N , X) is a Hilbert space. Let A ∈ L(l 2 ) have closed range. Then a particular generalized inverse, the Moore-Penrose inverse A + exists uniquely, and there are several ways to characterize it (see e.g. [5]): For example, A + is determined by the four Moore-Penrose equations Here A denotes the Hilbert space adjoint -which coincides with the formal adjoint Equivalently, A + is the desired Moore-Penrose inverse of A if and only if AA + is the orthogonal projection onto im A and I − A + A is the orthogonal Vol. 96 (2022) A note on generalized invertibility 649 projection onto ker A. Furthermore, it is also given by the following uniform limits They particularly simplify if A is left (or right) invertible: if ν(A) > 0 then (whenever x = 1) i.e. ν(A A) > 0. Together with (A A) = A A this yields invertibility. Thus, If A ∈ A l 2 is semi-Fredholm, i.e. has closed range and finite dimensional kernel (resp. cokernel) then C = A A (resp. C = AA ) is Fredholm and C + is a regularizer for C hence belongs to A l 2 by Remark 1. Equation (4)  In summary, if X is a Hilbert space, Theorem 1.1 for band-dominated operators on l 2 (Z N , X), as well as Theorems 1.2 and 1.3 for Wiener class operators on l p (Z N , X) remain true with the particular generalized inverse B = A + and without imposing the additional condition 'A is Fredholm' as in Sect. 4.1.

A further criterion for one-sided invertibility based on finite matrices
Here we finally return to the classical situation X = C once more. The previous formulas reveal a result which is already known from [3]: This opens the door for a subsequent study and a characterization of onesided invertibility in terms of finite matrices. Consider C = A A on l 2 and its compressions C n = P n CP n to the finite dimensional subspaces im P n . These C n can be regarded as finite square matrices acting on finite vectors with the Euclidean norm. Proof. From e.g. [12] it is well known that if these C n are invertible and their inverses have uniformly bounded norms (this is sometimes called stability of the sequence (C n )) then C is invertible. Actually, this can also be seen directly. Recall that Q n x → 0 on l 2 as n → ∞, assume that ν(C) = 0 and fix > 0. Choose x, x = 1 s.t. Cx ≤ and n so large that Q n x ≤ / C . Then, ν(C n ) ≤ C n x ≤ P n CP n x ≤ Cx + C Q n x ≤ 2 . Since was chosen arbitrarily this yields lim inf n ν(P n CP n ) = 0 contradicting the uniform invertibility. Thus, the self-adjoint C is left invertible hence invertible. Conversely, let C be invertible. Then for every x ∈ im R, x = 1, with R being an arbitrary self-adjoint projection, we have hence the restriction RCR is bounded below and due to its self-adjointness even invertible on im R with a bound on (RCR) −1 which is independent of R. Applying this to all R = P n we get the uniform invertibility of the operators C n .
The uniform invertibility of a sequence of matrices w.r.t. the norm · 2 , particularly if they are self-adjoint, can be characterized and checked by various tools. E.g. it means that there exists c > 0 such that all eigenvalues of all P n CP n are larger than c. In fact, as these matrices are positive, one only has Vol. 96 (2022) A note on generalized invertibility 651 to consider the smallest eigenvalues. This is also equivalent to having a c > 0 such that ν(P n CP n ) ≥ c for all n. Another equivalent characterization is the uniform boundedness of the condition numbers of all these matrices. Band-dominated operators A, and in particular A ∈ W, fulfill both P n AQ 2n → 0 and Q 2n AP n → 0 as n → ∞ (cf. e.g. [9, Theorem 2.1.6]). Hence P n A P 2n AP n − P n A AP n , P n AP 2n A P n − P n AA P n tend to 0, which with Proposition 4.7 and Corollary 4.6 gives the following characterization of one-sided invertibility: Theorem 4.8. Let A ∈ W. Then A is left (resp. right) invertible if and only if the sequence of positive matrices (P n A P 2n AP n ) (resp. (P n AP 2n A P n )) is uniformly invertible w.r.t. the Euclidean norm.
Unlike the criterion in [3,Theorem 3.14] the remarkable advantage of these matrices is that their construction only requires finitely many entries from the infinite matrix A, resp., hence their invertibility can effectively be checked by finite computations and e.g. one of the above mentioned tools for p = 2.
We point out that the results of this final Sect. 4.3 apply to A ∈ W on all l p with p ∈ {0} ∪ [1, ∞]. Moreover, they actually hold on l 2 with literally the same proofs for all band-dominated A in the superset A l 2 of W.
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