In this section, we discuss the normal form of continuous linear maps between locally convex spaces. Recall that every \((m \times n)\)-matrix T with rank r can be written in the following normal form
$$\begin{aligned} T = P \begin{pmatrix}0 &{} 0 \\ 0 &{} \mathbb {1}_{r \times r}\end{pmatrix} Q, \end{aligned}$$
(2.1)
where P and Q are invertible matrices of type \((m \times m)\) and \((n \times n)\), respectively. As we will see, a similar factorization is possible for continuous linear maps between locally convex spaces, which are relatively open and whose kernel and image are closed complemented subspaces. We call such operators regular and their associated representation (2.1) a normal form. As a preparation for the nonlinear case, we define and study regularity of families of linear maps depending continuously on a parameter. With a view toward applications, we give a brief overview of the theory of Fredholm operators and of elliptic operators in the locally convex framework and, in particular, show that these operators are regular.
Uniform regularity
Let X and Y be locally convex spaces, and let P be a neighborhood of 0 in some locally convex space. A continuous map \(T{\,\!:\;} P \times X \rightarrow Y\) is called a continuous family of linear maps if, for all \(p \in P\), the induced map \(T_p \equiv T(p, \cdot ){\,\!:\;} X \rightarrow Y\) is linear.
Definition 2.1
A continuous family \(T{\,\!:\;} P \times X \rightarrow Y\) of linear maps between locally convex spaces X and Y is called uniformly regular (at 0) if there exist topological decompositions
$$\begin{aligned} X = \mathop{\mathrm{Ker}}\, T_0 \oplus \mathop {\mathrm{Coim}}\, T_0, \qquad Y = \mathop{\mathrm {Coker}}\, T_0 \oplus \mathop{\mathrm {Im}}\, T_0, \end{aligned}$$
(2.2)
where \(\mathrm {Coim}\,T_0\) and \(\mathrm {Coker}\,T_0\) are closed subspacesFootnote 1 of X and Y, and, for every \(p \in P\), the map \(\tilde{T}_p = {\mathrm {pr}}_{\mathrm {Im}\,T_0} \circ \, (T_p)_{{\restriction }\mathrm {Coim}\,T_0}{\,\!:\;} \mathrm {Coim}\,T_0 \rightarrow \mathrm {Im}\,T_0\) is a topological isomorphism and the inverses \(\tilde{T}^{-1}_p\) form a continuous family \(P \times \mathrm {Im}\,T_0 \rightarrow \mathrm {Coim}\,T_0\) of isomorphisms.
In the case \(P = \{0\}\), the notion of uniform regularity reduces to the notion of a relatively open operator \(T{\,\!:\;} X \rightarrow Y\) with closed complemented kernel and image. We refer to this situation by saying that T is a regular operator.Footnote 2 Every regular operator T can be written in a normal form similar to (2.1):
$$\begin{aligned} T = P \begin{pmatrix}0 &{} 0 \\ 0 &{} \tilde{T}\end{pmatrix} Q, \end{aligned}$$
(2.3)
where \(Q{\,\!:\;} X \rightarrow \mathrm {Ker}\,T \oplus \mathrm {Coim}\,T\) and \(P{\,\!:\;}\mathrm {Coker}\,T \oplus \mathrm {Im}\,T \rightarrow Y\) are the natural isomorphisms determined by the decompositions (2.2), and \(\tilde{T}{\,\!:\;} \mathrm {Coim}\,T \rightarrow \mathrm {Im}\,T\) is a topological isomorphism. We call \(\tilde{T}\) (together with the isomorphisms Q and P) a normal form of T.
If the space of invertible maps is open in the space of all continuous linear maps, then, for every continuous family \(T{\,\!:\;} P \times X \rightarrow Y\) with \(T_0\) being regular, one can shrink P to pass to a uniformly regular family. This openness property fails when one leaves the Banach realm. However, when it does hold, uniform regularity reduces to a condition at one point.
Lemma 2.2
Let \(T{\,\!:\;} P \times X \rightarrow Y\) be a continuous family of linear maps between locally convex spaces X and Y. If \(\mathrm {Im}\,T_0\) is finite-dimensional, then T is uniformly regular after possibly shrinking P.
Proof
Since \(T_0\) has a finite-dimensional range, by [45, Proposition 20.5.5], \(\mathrm {Im}\,T_0\) is closed and has a topological complement. Moreover, \(\mathrm {Ker}\,T_0\) has finite codimension in X and hence is topologically complemented according to [45, Proposition 15.8.2]. The maps \(\tilde{T}_p{\,\!:\;}\mathrm {Coim}\,T_0 \rightarrow \mathrm {Im}\,T_0\) are continuous linear maps between finite-dimensional spaces. Since \(\tilde{T}_0\) is a bijection, the openness of the set of invertible operators implies that \(\tilde{T}_p\) is a topological isomorphism for \(p \in P\) close enough to 0. \(\square\)
Uniform regularity implies a semi-continuity property of the kernel and the image. Similar semi-continuity properties are well known for families of Fredholm operators between Banach spaces [42, Corollary 19.1.6].
Lemma 2.3
Let \(T{\,\!:\;} P \times X \rightarrow Y\) be a continuous family of linear maps between locally convex spaces X and Y. If T is uniformly regular, then the following holds:
-
1.
The kernel of T is upper semi-continuous at 0 in the sense that \(\mathrm {Ker}\,T_p \subseteq \mathrm {Ker}\,T_0\) for all \(p \in P\).
-
2.
The image of T is lower semi-continuous at 0 in the sense that \(\mathrm {Im}\,T_p \supseteq \mathrm {Im}\,T_0\) for all \(p \in P\). \(\square\)
Proof
The inclusions \(\mathrm {Ker}\,T_p \subseteq \mathrm {Ker}\,T_0\) and \(\mathrm {Im}\,T_p \supseteq \mathrm {Im}\,T_0\) need to be valid, because otherwise \(\tilde{T}_p = {\mathrm {pr}}_{\mathrm {Im}\,T_0} \circ \, (T_p)_{{\restriction }\mathrm {Coim}\,T_0}\) cannot be an isomorphism from \(\mathrm {Coim}\,T_0\) to \(\mathrm {Im}\,T_0\). \(\square\)
Uniform regularity is tightly connected to the invertibility of an extended operator.
Theorem 2.4
Let \(T{\,\!:\;} P \times X \rightarrow Y\) be a continuous family of linear maps between locally convex spaces X and Y. Then, the following are equivalent:
-
1.
T is uniformly regular.
-
2.
There exist locally convex spaces \(Z^\pm\), continuous linear maps \(T^+{\,\!:\;} Z^+ \rightarrow Y\) and \(T^-{\,\!:\;} X \rightarrow Z^-\), and continuous families of linear maps \(S{\,\!:\;} P \times Y \rightarrow X\), \(S^-{\,\!:\;} P \times Z^- \rightarrow X\), \(S^+{\,\!:\;} P \times Y \rightarrow Z^+\) and \(S^{-+}{\,\!:\;} P \times Z^- \rightarrow Z^+\) with \(S^{-+}_0 = 0\) such that
$$\begin{aligned} \begin{pmatrix}T_p &{} T^+ \\ T^- &{} 0\end{pmatrix}^{-1} = \begin{pmatrix}S_p &{} S^-_p \\ S^+_p &{} S^{-+}_p\end{pmatrix}, \end{aligned}$$
(2.4)
holds for all \(p \in P\) and such that the operators
$$\begin{aligned} \Gamma _p \equiv \begin{pmatrix} {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Coker}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, S^-_p \\ (S^+_p)_{{\restriction }\mathrm {Coker}\,T_0} &{} S^{-+}_p \end{pmatrix}{\,\!:\;} \mathrm {Coker}\,T_0 \oplus Z^- \rightarrow \mathrm {Ker}\,T_0 \oplus Z^+ \end{aligned}$$
(2.5)
are invertible for all \(p \in P\) and their inverses form a continuous family. \(\square\)
For the proof, we need the following basic result about the invertibility of block matrices in terms of the Schur complement.
Lemma 2.5
Let \(A_{11}{\,\!:\;} X_1 \rightarrow Y_1\), \(A_{12}{\,\!:\;} X_2 \rightarrow Y_1\), \(A_{21}{\,\!:\;} X_1 \rightarrow Y_2\) and \(A_{22}{\,\!:\;} X_2 \rightarrow Y_2\) be continuous linear maps between locally convex spaces such that
$$\begin{aligned} \begin{pmatrix}A_{11} &{} A_{12} \\ A_{21} &{} A_{22}\end{pmatrix}^{-1} = \begin{pmatrix}B_{11} &{} B_{12} \\ B_{21} &{} B_{22}\end{pmatrix} \end{aligned}$$
(2.6)
for continuous linear maps \(B_{ij}\) for \(i,j = 1,2\).
-
1.
If \(B_{22}\) is a topological isomorphism, then so is \(A_{11}\), and its inverse is given by
$$\begin{aligned} A_{11}^{-1} = B_{11} - B_{12} B_{22}^{-1} B_{21}. \end{aligned}$$
(2.7)
-
2.
If \(B_{22} = 0\), then \(A_{11} B_{11}\) and \(B_{11} A_{11}\) are idempotent and satisfy
$$\begin{aligned} \mathrm {Im}\,(A_{11} B_{11}) = \mathrm {Im}\,A_{11}, \qquad \quad \mathrm {Ker}\, (B_{11} A_{11}) = \mathrm {Ker}\,A_{11}. \end{aligned}$$
(2.8)
\(\square\)
Proof
The first statement is [75, Lemma 3.1] and can be verified by a direct calculation. The second statement follows from the identity
$$\begin{aligned} A_{11} B_{11} A_{11} = A_{11} (\mathrm {id}_{X_1} - B_{12}A_{21}) = A_{11}, \end{aligned}$$
(2.9)
where we used \(B_{11} A_{11} + B_{12} A_{21} = \mathrm {id}_{X_1}\) and \(A_{11} B_{12} = 0\). \(\square\)
Proof of Theorem 2.4
First, suppose that T is a uniformly regular family of linear maps. Then, by definition, we have topological decompositions \(X = \mathrm {Ker}\,T_0 \oplus \mathrm {Coim}\,T_0\), \(Y = \mathrm {Coker}\,T_0 \oplus \mathrm {Im}\,T_0\) and, for every \(p \in P\), the map
$$\begin{aligned} \tilde{T}_p = {\mathrm {pr}}_{\mathrm {Im}\,T_0} \circ \, (T_p)_{{\restriction }\mathrm {Coim}\,T_0} \end{aligned}$$
(2.10)
is a topological isomorphism. Set \(Z^+ = \mathrm {Coker}\,T_0\) and \(Z^- = \mathrm {Ker}\,T_0\), and let \(T^+{\,\!:\;} Z^+ \rightarrow Y\) and \(T^-{\,\!:\;} X \rightarrow Z^-\) be the canonical inclusion and projection, respectively. Moreover, let \(S_p{\,:}{=}\, \tilde{T}^{-1}_p\, \circ \, {\mathrm {pr}}_{\mathrm {Im}\,T_0}{\,\!:\;} Y \rightarrow X\), and define \(S^{-+}_p{\,\!:\;} Z^- \rightarrow Z^+\) by
$$\begin{aligned} S^{-+}_p = {\mathrm {pr}}_{\mathrm {Coker}\,T_0}\, \circ \, (T_p \,\circ \, \tilde{T}^{-1}_p\, \circ \, {\mathrm {pr}}_{\mathrm {Im}\,T_0} - \mathrm {id}_Y) \circ \, (T_p)_{{\restriction }\mathrm {Ker}\,T_0}\,. \end{aligned}$$
(2.11)
Finally, define \(S^\pm _p\) by
$$\begin{aligned} S^+_p&= {\mathrm {pr}}_{\mathrm {Coker}\,T_0} \, \circ \, (\mathrm {id}_Y - T_p \circ \, S_p){\,\!:\;} Y \rightarrow Z^+, \end{aligned}$$
(2.12a)
$$\begin{aligned} S^-_p&= (\mathrm {id}_X - S_p \, \circ \, T_p)_{{\restriction }\mathrm {Ker}\,T_0}{\,\!:\;} Z^- \rightarrow X. \end{aligned}$$
(2.12b)
Since, by definition, the inverses \(\tilde{T}^{-1}_p\) form a continuous family \(P \times \mathrm {Im}\,T_0 \rightarrow \mathrm {Coim}\,T_0\), the families \(S, S^\pm , S^{-+}\) are continuous. Furthermore, a direct calculation yields
$$\begin{aligned} T_p \, \circ \, S_p&= {\mathrm {pr}}_{\mathrm {Im}\,T_0} + {\mathrm {pr}}_{\mathrm {Coker}\,T_0} \circ \, T_p \circ \, \tilde{T}^{-1}_p \circ \, {\mathrm {pr}}_{\mathrm {Im}\,T_0}, \end{aligned}$$
(2.13a)
$$\begin{aligned} S_p \, \circ \, T_p&= {\mathrm {pr}}_{\mathrm {Coim}\,T_0} + \tilde{T}^{-1}_p\, \circ \, {\mathrm {pr}}_{\mathrm {Im}\,T_0} \circ \, (T_p)_{{\restriction }\mathrm {Ker}\,T_0}. \end{aligned}$$
(2.13b)
Using these identities, it is straightforward to check that (2.4) holds for every \(p \in P\). Moreover, we have
$$\begin{aligned} \Gamma _p = \begin{pmatrix} {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Coker}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, S^-_p \\ (S^+_p)_{{\restriction }\mathrm {Coker}\,T_0} &{} S^{-+}_p \end{pmatrix} = \begin{pmatrix} 0 &{} \mathrm {id}_{\mathrm {Ker}\,T_0} \\ \mathrm {id}_{\mathrm {Coker}\,T_0} &{} S^{-+}_p \end{pmatrix}, \end{aligned}$$
(2.14)
which is clearly invertible with a continuous family of inverses given by
$$\begin{aligned} \Gamma ^{\, -1}_p = \left( \begin{array}{cc} - S^{-+}_p &{} \mathrm {id}_{\mathrm {Coker}\,T_0} \\ \mathrm {id}_{\mathrm {Ker}\,T_0} &{} 0 \end{array}\right) . \end{aligned}$$
(2.15)
Conversely, let \(T^\pm , S, S^\pm\) and \(S^{-+}\) satisfying the assumptions of the second statement of Theorem 2.4. Since \(S^{-+}_0 = 0\), Lemma 2.5(2) implies that \(T_0\, \circ \, S_0\) and \(S_0\, \circ \, T_0\) are idempotent with \(\mathrm {Im}\,T_0 \,\circ \, S_0 = \mathrm {Im}\,T_0\) and \(\mathrm {Ker}\,S_0\, \circ \, T_0 = \mathrm {Ker}\,T_0\). Hence, \(\mathrm {Ker}\,T_0\) and \(\mathrm {Im}\,T_0\) are images of continuous idempotent operators, and as such they are closed and topologically complemented according to [45, Proposition 15.8.1]. As above, denote the complements by \(\mathrm {Coim}\,T_0\) and \(\mathrm {Coker}\,T_0\), respectively. It remains to show that \(\tilde{T}_p = {\mathrm {pr}}_{\mathrm {Im}\,T_0} \circ \, (T_p)_{{\restriction }\mathrm {Coim}\,T_0}\) is a topological isomorphism for all \(p \in P\), and that \(\tilde{T}_p^{-1}\) form a continuous family. For this purpose, we write all operators in block form with respect to the decompositions \(X = \mathrm {Coim}\,T_0 \oplus \mathrm {Ker}\,T_0\) and \(Y = \mathrm {Im}\,T_0 \oplus \mathrm {Coker}\,T_0\) (note the different order of the summands). Using this convention, the identity (2.4) becomes
$$\begin{aligned}&\begin{pmatrix} \tilde{T}_p &{} {\mathrm {pr}}_{\mathrm {Im}\,T_0} \circ \, (T_p)_{{\restriction }\mathrm {Ker}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Im}\,T_0} \circ \, T^+ \\ {\mathrm {pr}}_{\mathrm {Coker}\,T_0} \circ \, (T_p)_{{\restriction }\mathrm {Coim}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Coker}\,T_0} \circ \, (T_p)_{{\restriction }\mathrm {Ker}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Coker}\,T_0} \circ \, T^+ \\ (T^-)_{{\restriction }\mathrm {Coim}\,T_0} &{} (T^-)_{{\restriction }\mathrm {Ker}\,T_0} &{} 0 \end{pmatrix}^{-1} \nonumber \\&\qquad = \begin{pmatrix} {\mathrm {pr}}_{\mathrm {Coim}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Im}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Coim}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Coker}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Coim}\,T_0} \circ \, S^-_p \\ {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Im}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Coker}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, S^-_p \\ (S^+_p)_{{\restriction }\mathrm {Im}\,T_0} &{} (S^+_p)_{{\restriction }\mathrm {Coker}\,T_0} &{} S^{-+}_p\end{pmatrix}. \end{aligned}$$
(2.16)
These matrices should be read as operators from \(\mathrm {Im}\,T_0 \oplus \mathrm {Coker}\,T_0 \oplus Z^-\) to \(\mathrm {Coim}\,T_0 \oplus \mathrm {Ker}\,T_0 \oplus Z^+\). Note that the lower right two-times-two block of the right-hand side coincides with the operator \(\Gamma _p\). Since \(\Gamma _{p}\) is invertible by assumption, Lemma 2.5(1) shows that \(\tilde{T}_p\) is invertible, too. Moreover, the inverses are given by
$$\begin{aligned} \tilde{T}_p^{-1}&= {\mathrm {pr}}_{\mathrm {Coim}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Im}\,T_0} \nonumber \\&\quad - \begin{pmatrix}{\mathrm {pr}}_{\mathrm {Coim}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Coker}\,T_0} \\ {\mathrm {pr}}_{\mathrm {Coim}\,T_0} \circ \, S^-_p\end{pmatrix} \Gamma ^{\, -1}_{p} \begin{pmatrix}{\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Im}\,T_0} \\ (S^+_p)_{{\restriction }\mathrm {Im}\,T_0}\end{pmatrix} \end{aligned}$$
(2.17)
and thus form a continuous family \(P \times \mathrm {Im}\,T_0 \rightarrow \mathrm {Coim}\,T_0\). Hence, T is uniformly regular. \(\square\)
For the special case of a single operator, we obtain the following.
Corollary 2.6
A continuous linear map \(T{\,\!:\;} X \rightarrow Y\) between locally convex spaces is regular if and only if there exist locally convex spaces \(Z^\pm\), continuous linear maps \(T^+{\,\!:\;} Z^+ \rightarrow Y\), \(T^-{\,\!:\;} X \rightarrow Z^-\) and \(S{\,\!:\;} Y \rightarrow X\), \(S^-{\,\!:\;} Z^- \rightarrow X\), \(S^+{\,\!:\;} Y \rightarrow Z^+\) such that
$$\begin{aligned} \begin{pmatrix}T &{} T^+ \\ T^- &{} 0\end{pmatrix}^{-1} = \begin{pmatrix}S &{} S^- \\ S^+ &{} 0\end{pmatrix}. \end{aligned}$$
(2.18)
\(\square\)
Remark 2.7
In the setting of Corollary 2.6, it is straightforward to verify that \(T \, \circ \, S\, \circ \, T = T\) holds. An operator S satisfying such a relation is called a generalized inverse of T, cf. [56, 64, 66]. In fact, one can show that regularity of an operator is equivalent to the existence of a generalized inverse. Since we do not need this point of view in the remainder, we refer to [13, 36] for further details.
Remark 2.8
(Uniform regularity in the tame Fréchet category) It is clear that a version of Theorem 2.4 holds in the tame Fréchet category if the word “tame” is inserted in the right places (see Appendix A for a brief overview of the main concepts of tame Fréchet spaces). Let us spell out the details.
Let X and Y be tame Fréchet spaces and let \(T{\,\!:\;} P \times X \rightarrow Y\) be a tame smooth family of linear maps. Then, T is called uniformly tame regular if there exist tame decompositions \(X = \mathrm {Ker}\,T_0 \oplus \mathrm {Coim}\,T_0\) and \(Y = \mathrm {Coker}\,T_0 \oplus \mathrm {Im}\,T_0\), and, for every \(p \in P\), the map \(\tilde{T}_p = {\mathrm {pr}}_{\mathrm {Im}\,T_0} \circ \, (T_p)_{{\restriction }\mathrm {Coim}\,T_0}{\,\!:\;} \mathrm {Coim}\,T_0 \rightarrow \mathrm {Im}\,T_0\) is a tame isomorphism such that the inverses form a tame smooth family \(P \times \mathrm {Im}\,T_0 \rightarrow \mathrm {Coim}\,T_0\). Then, the equivalence of Theorem 2.4 holds with \(Z^\pm\) being tame Fréchet spaces, \(T^\pm\) being tame maps and \(S, S^\pm , S^{-+}, \Gamma ^{-1}\) being tame smooth families.
Fredholm operators
An important class of examples of regular operators is given by Fredholm operators. Fredholm operators are usually studied as maps between Banach spaces (or Hilbert spaces), but most results extend to the locally convex setting, cf. [22, 71, 72].
Definition 2.9
A continuous linear map \(T{\,\!:\;} X \rightarrow Y\) between locally convex spaces is called a Fredholm operator if T is relatively open, the kernel of T is a finite-dimensional subspace of X, and the image of T is a finite-codimensional closed subspace of Y. The index \(\mathrm {ind}\,T\) of a Fredholm operator T is defined by
$$\begin{aligned} \mathrm {ind}\,T = \dim \mathrm {Ker}\,T - \dim \mathrm {Coker}\,T. \end{aligned}$$
(2.19)
\(\square\)
Since finite-dimensional subspaces and finite-codimensional closed subspaces of a locally convex space are always topologically complemented according to [45, Propositions 15.8.2 and 20.5.5], every Fredholm operator is regular.
For a continuous family T of linear maps with \(T_0\) being a Fredholm operator, the invertibility of the family \(\Gamma\) in Theorem 2.4 is automatic.
Corollary 2.10
Let \(T{\,\!:\;} P \times X \rightarrow Y\) be a continuous family of linear maps between locally convex spaces X and Y such that \(T_0\) is a Fredholm operator. Then, T is uniformly regular if and only if there exist finite-dimensional spaces \(Z^\pm\) and continuous linear maps \(T^+{\,\!:\;} Z^+ \rightarrow Y\) and \(T^-{\,\!:\;} X \rightarrow Z^-\) such that, after possibly shrinking P,
$$\begin{aligned} \begin{pmatrix}T_p &{} T^+ \\ T^- &{} 0\end{pmatrix}^{-1} = \begin{pmatrix}S_p &{} S^-_p \\ S^+_p &{} S^{-+}_p\end{pmatrix} \end{aligned}$$
(2.20)
holds for all \(p \in P\), where \(S{\,\!:\;} P \times Y \rightarrow X\), \(S^-{\,\!:\;} P \times Z^- \rightarrow X\), \(S^+{\,\!:\;} P \times Y \rightarrow Z^+\) and \(S^{-+}{\,\!:\;} P \times Z^- \rightarrow Z^+\) are continuous families of linear maps with \(S^{-+}_0 = 0\).
Proof
If T is uniformly regular, then the proof of Theorem 2.4 shows that one can choose \(Z^+ = \mathrm {Coker}\,T_0\) and \(Z^- = \mathrm {Ker}\,T_0\). Both spaces are finite-dimensional, because \(T_0\) is a Fredholm operator. This establishes one direction.
Conversely, let \(T^\pm , S, S^\pm , S^{-+}\) be given as stated above. By Theorem 2.4, it suffices to show that the operator
$$\begin{aligned} \Gamma _p = \begin{pmatrix} {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, (S_p)_{{\restriction }\mathrm {Coker}\,T_0} &{} {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, S^-_p \\ (S^+_p)_{{\restriction }\mathrm {Coker}\,T_0} &{} S^{-+}_p \end{pmatrix}{\,\!:\;} \mathrm {Coker}\,T_0 \oplus Z^- \rightarrow \mathrm {Ker}\,T_0 \oplus Z^+ \end{aligned}$$
(2.21)
is invertible and that the inverses form a continuous family. Since \(S^{-+}_0 = 0\), Lemma 2.5(2) implies \(S^-_0 \, \circ \, T^- = {\mathrm {pr}}_{\mathrm {Ker}\,T_0}\) and \(T^+ \, \circ \, S^+_0 = {\mathrm {pr}}_{\mathrm {Coker}\,T_0}\). A straightforward calculation using these identities shows that we have
$$\begin{aligned} \Gamma _0 = \begin{pmatrix} 0 &{} {\mathrm {pr}}_{\mathrm {Ker}\,T_0} \circ \, S^-_0 \\ (S^+_0)_{{\restriction }\mathrm {Coker}\,T_0} &{} 0 \end{pmatrix}, \end{aligned}$$
(2.22)
and
$$\begin{aligned} \Gamma _0^{\, -1} = \begin{pmatrix} 0 &{} {\mathrm {pr}}_{\mathrm {Coker}\,T_0} \circ \, T^+ \\ (T^-)_{{\restriction }\mathrm {Ker}\,T_0} &{} 0 \end{pmatrix}. \end{aligned}$$
(2.23)
Since, for every \(p \in P\), \(\Gamma _p\) is an operator between finite-dimensional spaces and \(\Gamma _0\) is invertible, we can shrink P in such a way that \(\Gamma _p\) is invertible for all \(p \in P\) and that the inverses form a continuous family. Thus, Theorem 2.4 implies that T is uniformly regular. \(\square\)
In the Banach setting, the set of Fredholm operators is open in the space of bounded linear operators and the index does not change under a continuous deformation [42, Corollary 19.1.6]. This statement relies on the openness of the set of invertible operators and thus does not carry over to the locally convex setting. The following proposition shows that the notion of uniform regularity is an adequate substitute.
Proposition 2.11
Let \(T{\,\!:\;} P \times X \rightarrow Y\) be a uniformly regular family of linear maps between locally convex spaces X and Y. If \(T_0\) is Fredholm, then \(T_p\) is Fredholm and \(\mathrm {ind}\,T_p = \mathrm {ind}\,T_0\) for all \(p \in P\).
Proof
By Corollary 2.10, there exist finite-dimensional spaces \(Z^\pm\) and continuous linear maps \(T^+{\,\!:\;} Z^+ \rightarrow Y\) and \(T^-{\,\!:\;} X \rightarrow Z^-\) such that
$$\begin{aligned} \begin{pmatrix}T_p &{} T^+ \\ T^- &{} 0\end{pmatrix} \end{aligned}$$
(2.24)
is invertible for all \(p \in P\). This is only possible if \(\mathrm {Ker}\,T_p\) and \(\mathrm {Coker}\,T_p\) are finite-dimensional so that \(T_p\) has to be Fredholm. Moreover, using the invariance of the index under finite-rank perturbations, we have
$$\begin{aligned} 0 = \mathrm {ind}\,\begin{pmatrix} T_p &{} T^+ \\ T^- &{} 0\end{pmatrix} = \mathrm {ind}\,T_p + \dim Z^+ - \dim Z^- = \mathrm {ind}\,T_p - \mathrm {ind}\,T_0, \end{aligned}$$
(2.25)
which establishes the formula for the index. \(\square\)
As we discuss now, families of elliptic operators constitute an important class of examples of uniformly regular Fredholm operators. Let \(E \rightarrow M\) and \(F \rightarrow M\) be finite-dimensional vector bundles over a compact manifold M without boundary. Endow the spaces \(\mathcal {E}\) and \(\mathcal {F}\) of smooth sections of E and F, respectively, with the compact-open \(\text {C}^{\infty }\)-topology. With respect to this topology, these section spaces are tame Fréchet spaces, see [35, Theorem II.2.3.1]. A continuous linear map \(L{\,\!:\;} \mathcal {E} \rightarrow \mathcal {F}\) is a partial differential operator of degree r if and only if there exists a vertical vector bundle morphism \(f{\,\!:\;} \mathrm {J}^r E \rightarrow F\) such that L factors through the jet bundle \(\mathrm {J}^r E\) as follows:
where \(\mathrm {j}^r\) denotes the r-th jet prolongation and \(f_*\) is the push-forward by f. We refer to f as the coefficients of L and will sometimes write \(L_f\) instead of L to emphasize this relation. Recall that the principal symbol \(\sigma _f\) of \(L_f\) is a homogeneous polynomial of degree r on \(\mathrm {T}^{*}M\) with values in the bundle \(\mathrm {L}(E, F)\) of fiberwise linear maps \(E \rightarrow F\). A differential operator \(L_f\) with coefficients f is called elliptic if its symbol is invertible; that is, for each nonzero \(p \in \mathrm {T}^{*}M\), the bundle map \(\sigma _f (p, \dotsc , p) \in \mathrm {L}(E, F)\) is invertible.
It is a standard result in elliptic theory that every elliptic differential operator over a compact manifold is a Fredholm operator between appropriate Sobolev spaces [42, Theorem 19.2.1]. The same holds true in the tame Fréchet category. In fact, more is true: elliptic operators are regularly parametrized by their coefficients.
Theorem 2.12
Let \(E \rightarrow M\) and \(F \rightarrow M\) be finite-dimensional vector bundles over a compact manifold M without boundary, and denote the space of smooth sections of E and F by \(\mathcal {E}\) and \(\mathcal {F}\), respectively. The parametrization of a partial differential operator by their coefficients yields a tame smooth family
$$\begin{aligned} L{\,\!:\;} {\Gamma ^{\infty }\bigl (\mathrm {L}(\mathrm {J}^r E, F)\bigl )} \times \mathcal {E} \rightarrow \mathcal {F}, \quad (f, \phi ) \mapsto L_f (\phi ) \end{aligned}$$
(2.27)
of linear operators which is uniformly tame regular in a neighborhood of every \(f_0 \in \Gamma ^{\infty }\bigl (\mathrm {L}(\mathrm {J}^r E, F)\bigl )\) for which \(L_{f_0}\) is an elliptic differential operator.
Proof
Let \(f_0 \in \Gamma ^{\infty }\bigl (\mathrm {L}(\mathrm {J}^r E, F)\bigl )\) be such that \(L_{f_0}\) is an elliptic differential operator. By [35, Theorem II.3.3.3], there exist an open neighborhood \(\mathcal {U}\) of \(f_0\) in \(\Gamma ^{\infty }\bigl (\mathrm {L}(\mathrm {J}^r E, F)\bigl )\), finite-dimensional vector spaces \(Z^\pm\) and continuous linear maps \(L^+{\,\!:\;} Z^+ \rightarrow Y\) and \(L^-{\,\!:\;}X \rightarrow Z^-\) such that
$$\begin{aligned} \begin{pmatrix}L_f &{} L^+ \\ L^- &{} 0\end{pmatrix}{\,\!:\;} \mathcal {E} \times Z^+ \rightarrow \mathcal {F} \times Z^- \end{aligned}$$
(2.28)
is invertible for all \(f \in \mathcal {U}\). Moreover, the inverses form a tame smooth family \(\mathcal {U} \times \mathcal {F} \times Z^- \rightarrow \mathcal {E} \times Z^+\) of linear operators. Hence, by Corollary 2.10, \(L_{{\restriction }\mathcal {U} \times \mathcal {E} }\) is uniformly tame regular at \(f_0\). \(\square\)
Elliptic complexes
In this section, the notion of uniform regularity is extended to linear chain complexes. The main application we have in mind is elliptic complexes.
Let P be a neighborhood of 0 in some locally convex space, let \(X_i\) be a sequence of locally convex spaces and let \(T_i{\,\!:\;} P \times X_i \rightarrow X_{i+1}\) be a sequence of continuous families of linear maps such that \((X_i, T_{i, 0})\) is a complex, i.e., \(T_{i+1, 0} \,\circ \, T_{i, 0} = 0\) for all \(i \in \mathbb {Z}\). We say that \((P, X_i, T_{i})\) is a continuous family of chains. Simple examples (cf. Example 2.16 below) show that a deformation of a chain complex is in general not a complex; this is why we require \(T_{i, p}\) to form a complex only at \(p = 0\). The following notion is a natural generalization of uniform regularity to chains.
Definition 2.13
A continuous family of chains \((P, X_i, T_{i})\) is called uniformly regular (at 0) if the following holds for every \(i \in \mathbb {Z}\):
-
1.
The image of \(T_{i-1, 0}\) is closed in \(X_i\), and there exist closed subspaces \(H_i\) and \(\mathrm {Coim}\,T_{i, 0}\) of \(X_i\) such that
$$\begin{aligned} X_i = \mathrm {Im}\,T_{i-1, 0} \oplus \mathrm {Coim}\,T_{i, 0} \oplus H_i \end{aligned}$$
(2.29)
is a topological decomposition and \(H_i \subseteq \mathrm {Ker}\,T_{i,0}\).
-
2.
For every \(p \in P\), the map
$$\begin{aligned} \tilde{T}_{i, p} = {\mathrm {pr}}_{\mathrm {Im}\,T_{i, 0}} \circ \, (T_{i, p})_{{\restriction }\mathrm {Coim}\,T_{i, 0}}{\,\!:\;} \mathrm {Coim}\,T_{i, 0} \rightarrow \mathrm {Im}\,T_{i, 0} \end{aligned}$$
(2.30)
is a topological isomorphism such that the inverses form a continuous family \(P \times \mathrm {Im}\,T_{i, 0} \rightarrow \mathrm {Coim}\,T_{i,0}\).
If additionally, for every \(i \in \mathbb {Z}\), \(X_i\) is a tame Fréchet space, \(T_i\) is a tame smooth family, the decomposition (2.29) of \(X_i\) is tame and \(\tilde{T}_{i, p}\) are a tame isomorphisms such that the inverses form a tame smooth family, then \((P, X_i, T_i)\) is called uniformly tame regular.
By definition, for an uniformly regular family \((P, X_i, T_{i})\) of chains, we have
$$\begin{aligned} \mathrm {Ker}\,T_{i, 0} = \mathrm {Im}\,T_{i-1,0} \oplus H_i \, , \end{aligned}$$
(2.31)
which justifies the notion \(\mathrm {Coim}\,T_{i,0}\) for the subspace in the decomposition (2.29). The subspaces \(H_i\) are identified with the homology groups for the complex at \(p = 0\), that is,
$$\begin{aligned} H_i \simeq \mathrm {Ker}\,T_{i, 0} /\mathrm {Im}\,T_{i-1, 0}. \end{aligned}$$
(2.32)
For the applications we have in mind, the following characterization of uniform regularity of chains turns out to be more convenient. It entails that, roughly speaking, a family of chains \((P, X_i, T_{i})\) is uniformly regular if each family \(T_i\) of linear maps is uniformly regular after factoring-out the image of the direct predecessor \(T_{i-1, 0}\).
Proposition 2.14
A continuous family of chains \((P, X_i, T_{i})\) is uniformly regular if and only if, for every \(i \in \mathbb {Z}\), the subspace \(\mathrm {Im}\,T_{i-1, 0}\) of \(X_i\) is closed and topologically complemented, say \(X_i = \mathrm {Im}\,T_{i-1, 0} \oplus \mathrm {Coker}\,T_{i-1, 0}\), and the continuous family \(p \mapsto (T_{i, p})_{{\restriction }\mathrm {Coker}\,T_{i-1, 0}}\) of linear maps is uniformly regular.
Proof
The claim is a simple consequence of the observation that the image of \((T_{i, 0})_{{\restriction }\mathrm {Coker}\,T_{i-1, 0}}\) coincides with the image of \(T_{i, 0}\) and that
$$\begin{aligned} H_i \simeq \mathrm {Ker}\,{(T_{i, 0})_{{\restriction }\mathrm {Coker}\,T_{i-1, 0}}} \end{aligned}$$
(2.33)
holds, because \(T_{i,0}\) is a complex. \(\square\)
Let us now turn to deformations of elliptic complexes. Let \(E_0, E_1, \dotsc , E_N\) be a sequence of finite-dimensional vector bundles over a compact manifold M, and let \(\mathcal {E}_i\) be the tame Fréchet space of smooth sections of \(E_i\). Moreover, let P be an open neighborhood of 0 in some tame Fréchet space and let \(L_{i}{\,\!:\;} P \times \mathcal {E}_i \rightarrow \mathcal {E}_{i+1}\) be a sequence of differential operators parametrized by points of P. We assume that, for every \(i \in \mathbb {Z}\), the parametrization factors through the space of coefficients as follows:
where \(\hat{L}_i{\,\!:\;} P \to \Gamma ^{\infty }\bigl (\mathrm {L}(\mathrm {J}^{r_i} E_i, E_{i+1})\bigl )\) is a tame smooth map and the second map is the parametrization of differential operators by their coefficients as defined in (2.27). For simplicity, let us assume that the degree \(r_i\) of the differential operator \(L_{i, p}{\,\!:\;} \mathcal {E}_i \rightarrow \mathcal {E}_{i+1}\) is the same for all \(p \in P\) and \(i \in \mathbb {Z}\). We will refer to this setting by saying that \((P, \mathcal {E}_i, L_{i})\) is a tame family of chains of differential operators. A chain complex \(L_i{\,\!:\;} \mathcal {E}_i \rightarrow \mathcal {E}_{i+1}\) of differential operators is called elliptic if the sequence of principal symbols
is exact outside of the zero section of the cotangent bundle \({\mathop {\tau }\limits ^{\star }}{\,\!:\;} \mathrm {T}^{*}M \rightarrow M\).
As a generalization of Theorem 2.12, we have the following result concerning deformations of elliptic complexes.
Theorem 2.15
Let \(E_0, \dotsc , E_N\) be a sequence of finite-dimensional vector bundles over a compact manifold M, and denote the space of smooth sections of \(E_i\) by \(\mathcal {E}_i\). Let \((P, \mathcal {E}_i, L_{i})\) be a tame family of chains of differential operators. If \((\mathcal {E}_i, L_{i, 0})\) is an elliptic complex, then \((P, \mathcal {E}_i, L_{i})\) is uniformly tame regular (after possibly shrinking P).
Proof
The proof is inspired by the proof of [3, Proposition 6.1], where a parametrix of an elliptic complex is constructed by using the parametrix of an elliptic operator. Similarly, we will reduce the question of the uniform tame regularity of the chain to the uniform regularity of a deformation of differential operators, for which we can employ Theorem 2.12.
For this purpose, fix a Riemannian metric on M and a fiber Riemannian metric on every vector bundle \(E_i\). These data define a natural \(\mathrm {L}^{2}\)-inner product on \(\mathcal {E}_i\). By partial integration, we see that the adjoints \(L^*_{i, p}{\,\!:\;}\mathcal {E}_{i+1} \rightarrow \mathcal {E}_i\) of \(L_{i,p}\) with respect to these inner products yield a tame family of chains of differential operators. For every \(i \in \mathbb {Z}\), define the tame family \(\Delta _{i}{\,\!:\;}P \times \mathcal {E}_i \rightarrow \mathcal {E}_i\) by
$$\begin{aligned} \Delta _{i, p} = L_{i,0}^* \, \circ \, L_{i,p} + L_{i-1,p} \, \circ \, L_{i-1,0}^*. \end{aligned}$$
(2.36)
Clearly, \(\Delta _{i}\) is a family of differential operators of order 2r. Moreover, \(\Delta _{i, 0}\) is an elliptic operator, because \((\mathcal {E}_i, L_{i, 0})\) is an elliptic complex by assumption. Thus, Theorem 2.12 implies that the family \(\Delta _{i}\) is uniformly tame regular. In particular, \(\Delta _{i, 0}\) is regular and self-adjoint so that we get the following topological decomposition
$$\begin{aligned} \mathcal {E}_i = \mathrm {Ker}\,\Delta _{i, 0} \oplus \mathrm {Im}\,\Delta _{i, 0} \, . \end{aligned}$$
(2.37)
Moreover, \(\tilde{\Delta }_{i, p} = {\mathrm {pr}}_{\mathrm {Im}\,\Delta _{i, 0}} \circ \, (\Delta _{i, p})_{{\restriction }\mathrm {Im}\,\Delta _{i, 0}}\) is a tame automorphism of \(\mathrm {Im}\,\Delta _{i, 0}\) for every \(p \in P\) (after possibly shrinking P) in such a way that the inverses form a tame smooth family. The decomposition (2.37) implies that the images of \(L_{i-1,0}\) and \(L^*_{i,0}\) are closed and that they fit into the topological decomposition
$$\begin{aligned} \mathcal {E}_i = \mathrm {Im}\,L_{i-1,0} \oplus \mathrm {Im}\,L^*_{i,0} \oplus H_i \, , \end{aligned}$$
(2.38)
where \(H_i \equiv \mathrm {Ker}\,\Delta _{i, 0} = \mathrm {Ker}\,L_{i,0} \cap \mathrm {Ker}\,L^*_{i-1, 0}\). Finally, a direct calculation shows that, for every \(i \in \mathbb {Z}\), the tame smooth family \(G_i\) defined by
$$\begin{aligned} G_{i,p} = L^*_{i, 0} \, \circ \, (\tilde{\Delta }_{i+1,p}^{-1})_{{\restriction }\mathrm {Im}\,L_{i, 0}}{\,\!:\;} \mathrm {Im}\,L_{i, 0} \rightarrow \mathrm {Im}\,L^*_{i, 0} \end{aligned}$$
(2.39)
is an inverse of the family
$$\begin{aligned} \tilde{L}_{i, p} = {\mathrm {pr}}_{\mathrm {Im}\,L_{i, 0}} {\, \circ \,} (L_{i, p})_{{\restriction }\mathrm {Im}\,L^*_{i, 0}}{\,\!:\;} \mathrm {Im}\,L^*_{i, 0} \rightarrow \mathrm {Im}\,L_{i, 0}. \end{aligned}$$
(2.40)
This shows that \((P, \mathcal {E}_i, L_{i})\) is uniformly tame regular. \(\square\)
Example 2.16
Let \(P \rightarrow M\) be a finite-dimensional principal G-bundle over a compact manifold M, and let E be an associated vector bundle. The space \(\mathcal {C}(P)\) of connections on P is an affine tame Fréchet space. Every connection \(A \in \mathcal {C}(P)\) yields via the covariant exterior differential on E-valued forms a chain
As this chain is an elliptic complex if the connection is flat, Theorem 2.15 entails that the family of chains \(\bigl (\mathcal {C}(P), \Omega ^k(M, E), \mathrm {d}_A \bigr )\) is uniformly tame regular in a neighborhood of every flat connection \(A_0 \in \mathcal {C}(P)\). Moreover, by Proposition 2.14, the family
$$\begin{aligned} \mathcal {C}(P) \times \Omega ^0(M, E) \rightarrow \Omega ^{1}(M, E), \qquad (A, \alpha ) \mapsto \mathrm {d}_A \alpha \end{aligned}$$
(2.42)
is uniformly regular at every \(A_0 \in \mathcal {C}(P)\) (we do not need flatness of \(A_0\) for this in 0-degree). The operator defined in (2.36) takes here the following form
$$\begin{aligned} \Delta _{A_0 A} = \mathrm {d}^*_{A_0} \mathrm {d}_A + \mathrm {d}^*_A \mathrm {d}_{A_0}{\,\!:\;} \Omega ^{k}(M, E) \rightarrow \Omega ^{k}(M, E) \end{aligned}$$
(2.43)
and is a natural extension of the Faddeev–Popov operator to forms of higher degree, cf. [70, eq. (8.4.8)]. A similar operator played a central role in [15, p. 405] for the study of the curvature map \(F{\,\!:\;} \mathcal {C}(P) \rightarrow \Omega ^2(M, \mathrm {Ad}P)\) near a flat connection.