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The following is a well-known result for elliptic operators.

Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact manifold $X$. Then $P$ extends to a Fredholm map $P: W^{k,2}(E) \to W^{k-m,2}(F)$ whose index is independent of $k$ (cf. p193. Lawson-Michelsohn's book 'Spin Geometry')

This kind of extension involves Hilbert spaces and thus is relatively not hard to work with. However, concerning Floer theory (or pseudoholomorphic curvers theory), the linearized maps we concern have to be between non-Hilbert Banach spaces, such as $W^{1,p}(\mathbb R \times S^1, u^*TM)$ or $L^p(S^2, \Lambda^{0,1} \otimes_J u^*TM)$, where $p$ is always assumed to be greater than 2.

In the case $p\neq 2$, it seems much harder to show any Fredholm properties, and the proofs are usually not natural but very technical. You may agree to me if you refer to Audin-Damian's book(section 8.7) or McDuff-Salamon's book(Proposition 3.1.11 which requires Theorem C.2.3) respectively.

So, I would like to ask does there exist any general theorem concerning Fredholm properties, which is similar to the above one? If exist, what could be a general principle to show the Fredholm properties? In the above theorem, can we relax the condition that the base manifold $X$ is compact, as in Floer theory, the base manifold $\mathbb R \times S^1$ is non-compact?

Edit: I recently found a paper (cf. (0.1) in the very beginning) which states that (if I haven't misunderstood): let $E$ and $F$ be two vector bundles over a compact manifold $X$ and let $P: \Gamma(E)\to \Gamma(F)$ is an elliptic differential operator of order $m$, then the paper claims that it is well-known that it has a Fredholm extension $$ P:W^{k+m,p}(E) \to W^{k,p}(F)$$ for every $1<p<\infty$. This should be a really good result but I have trouble in finding a serious reference to confirm this.

The following is a well-known result for elliptic operators.

Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact manifold $X$. Then $P$ extends to a Fredholm map $P: W^{k,2}(E) \to W^{k-m,2}(F)$ whose index is independent of $k$ (cf. p193. Lawson-Michelsohn's book 'Spin Geometry')

This kind of extension involves Hilbert spaces and thus is relatively not hard to work with. However, concerning Floer theory (or pseudoholomorphic curvers theory), the linearized maps we concern have to be between non-Hilbert Banach spaces, such as $W^{1,p}(\mathbb R \times S^1, u^*TM)$ or $L^p(S^2, \Lambda^{0,1} \otimes_J u^*TM)$, where $p$ is always assumed to be greater than 2.

In the case $p\neq 2$, it seems much harder to show any Fredholm properties, and the proofs are usually not natural but very technical. You may agree to me if you refer to Audin-Damian's book(section 8.7) or McDuff-Salamon's book(Proposition 3.1.11 which requires Theorem C.2.3) respectively.

So, I would like to ask does there exist any general theorem concerning Fredholm properties, which is similar to the above one? If exist, what could be a general principle to show the Fredholm properties? In the above theorem, can we relax the condition that the base manifold $X$ is compact, as in Floer theory, the base manifold $\mathbb R \times S^1$ is non-compact? Any great reference?

Edit: I recently found a paper (cf. (0.1) in the very beginning) which states that (if I haven't misunderstood): let $E$ and $F$ be two vector bundles over a compact manifold $X$ and let $P: \Gamma(E)\to \Gamma(F)$ is an elliptic differential operator of order $m$, then the paper claims that it is well-known that it has a Fredholm extension $$ P:W^{k+m,p}(E) \to W^{k,p}(F)$$ for every $1<p<\infty$. This should be a really good result but I have trouble in finding a serious reference to confirm this.

The following is a well-known result for elliptic operators.

Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact manifold $X$. Then $P$ extends to a Fredholm map $P: W^{k,2}(E) \to W^{k-m,2}(F)$ whose index is independent of $k$ (cf. p193. Lawson-Michelsohn's book 'Spin Geometry')

This kind of extension involves Hilbert spaces and thus is relatively not hard to work with. However, concerning Floer theory (or pseudoholomorphic curvers theory), the linearized maps we concern have to be between non-Hilbert Banach spaces, such as $W^{1,p}(\mathbb R \times S^1, u^*TM)$ or $L^p(S^2, \Lambda^{0,1} \otimes_J u^*TM)$, where $p$ is always assumed to be greater than 2.

In the case $p\neq 2$, it seems much harder to show any Fredholm properties, and the proofs are usually not natural but very technical. You may agree to me if you refer to Audin-Damian's book(section 8.7) or McDuff-Salamon's book(Proposition 3.1.11 which requires Theorem C.2.3) respectively.

So, I would like to ask does there exist any general theorem concerning Fredholm properties, which is similar to the above one? If exist, what could be a general principle to show the Fredholm properties?

PS: In the above theorem, can we relax the condition that the base manifold $X$ is compact, as in Floer theory, the base manifold $\mathbb R \times S^1$ is non-compact?

The following is a well-known result for elliptic operators.

Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact manifold $X$. Then $P$ extends to a Fredholm map $P: W^{k,2}(E) \to W^{k-m,2}(F)$ whose index is independent of $k$ (cf. p193. Lawson-Michelsohn's book 'Spin Geometry')

This kind of extension involves Hilbert spaces and thus is relatively not hard to work with. However, concerning Floer theory (or pseudoholomorphic curvers theory), the linearized maps we concern have to be between non-Hilbert Banach spaces, such as $W^{1,p}(\mathbb R \times S^1, u^*TM)$ or $L^p(S^2, \Lambda^{0,1} \otimes_J u^*TM)$, where $p$ is always assumed to be greater than 2.

In the case $p\neq 2$, it seems much harder to show any Fredholm properties, and the proofs are usually not natural but very technical. You may agree to me if you refer to Audin-Damian's book(section 8.7) or McDuff-Salamon's book(Proposition 3.1.11 which requires Theorem C.2.3) respectively.

So, I would like to ask does there exist any general theorem concerning Fredholm properties, which is similar to the above one? If exist, what could be a general principle to show the Fredholm properties? In the above theorem, can we relax the condition that the base manifold $X$ is compact, as in Floer theory, the base manifold $\mathbb R \times S^1$ is non-compact?

Edit: I recently found a paper (cf. (0.1) in the very beginning) which states that (if I haven't misunderstood): let $E$ and $F$ be two vector bundles over a compact manifold $X$ and let $P: \Gamma(E)\to \Gamma(F)$ is an elliptic differential operator of order $m$, then the paper claims that it is well-known that it has a Fredholm extension $$ P:W^{k+m,p}(E) \to W^{k,p}(F)$$ for every $1<p<\infty$. This should be a really good result but I have trouble in finding a serious reference to confirm this.

The following is a well-known result for elliptic operators.

Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact manifold $X$. Then $P$ extends to a Fredholm map $P: W^{k,2}(E) \to W^{k-m,2}(F)$ whose index is independent of $k$ (cf. p193. Lawson-Michelsohn's book 'Spin Geometry')

This kind of extension involves Hilbert spaces and thus is relatively not hard to work with. However, concerning Floer theory (or pseudoholomorphic curvers theory), the linearized maps we concern have to be between non-Hilbert Banach spaces, such as $W^{1,p}(\mathbb R \times S^1, u^*TM)$ or $L^p(S^2, \Lambda^{0,1} \otimes_J u^*TM)$, where $p$ is always assumed to be greater than 2.

In the case $p\neq 2$, it seems much harder to show any Fredholm properties, and the proofs are usually not natural but very technical. You may agree to me if you refer to Audin-Damian's book(section 8.7) or McDuff-Salamon's book(Proposition 3.1.11 which requires Theorem C.2.3) respectively.

So, I would like to ask does there exist any general theorem concerning Fredholm properties, which is similar to the above one? If exist, what could be a general principle to show the Fredholm properties?

PS: In the above theorem, can we relax the condition that the base manifold $X$ is compact?

The following is a well-known result for elliptic operators.

Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact manifold $X$. Then $P$ extends to a Fredholm map $P: W^{k,2}(E) \to W^{k-m,2}(F)$ whose index is independent of $k$ (cf. p193. Lawson-Michelsohn's book 'Spin Geometry')

This kind of extension involves Hilbert spaces and thus is relatively not hard to work with. However, concerning Floer theory (or pseudoholomorphic curvers theory), the linearized maps we concern have to be between non-Hilbert Banach spaces, such as $W^{1,p}(\mathbb R \times S^1, u^*TM)$ or $L^p(S^2, \Lambda^{0,1} \otimes_J u^*TM)$, where $p$ is always assumed to be greater than 2.

In the case $p\neq 2$, it seems much harder to show any Fredholm properties, and the proofs are usually not natural but very technical. You may agree to me if you refer to Audin-Damian's book(section 8.7) or McDuff-Salamon's book(Proposition 3.1.11 which requires Theorem C.2.3) respectively.

So, I would like to ask does there exist any general theorem concerning Fredholm properties, which is similar to the above one? If exist, what could be a general principle to show the Fredholm properties?

PS: In the above theorem, can we relax the condition that the base manifold $X$ is compact, as in Floer theory, the base manifold $\mathbb R \times S^1$ is non-compact?