Normally-solvable operator

A linear operator with closed range. Let $A$ be a linear operator with dense domain in a Banach space $X$ and with range $R ( A)$ in a Banach space $Y$. Then $A$ is normally solvable if $\overline{ {R( A) }}\; = R ( A)$, that is, if $R ( A)$ is a closed subspace of $Y$. Let $A ^ {*}$ be the adjoint of $A$. For $A$ to be normally solvable it is necessary and sufficient that $R ( A) = ^ \perp N ( A ^ {*} )$, that is, that the range of $A$ is the orthogonal complement to the null space of $A ^ {*}$.

Suppose that

$$\tag{* } A x = y$$

is an equation with a normally-solvable operator (a normally-solvable equation). If $N ( A ^ {*} ) = \{ 0 \}$, that is, if the homogeneous adjoint equation $A ^ {*} \psi = 0$ has only the trivial solution, then $R ( A) = Y$. But if $N ( A ^ {*} ) \neq \{ 0 \}$, then for (*) to be solvable it is necessary and sufficient that $\langle y , \psi \rangle = 0$ for all solutions of the equation $A ^ {*} \psi = 0$.

From now on suppose that $A$ is closed. A normally-solvable operator is called $n$- normal if its null space $N ( A)$ is finite dimensional $( n( A) = \mathop{\rm dim} N ( A) < + \infty )$. A normally-solvable operator $A$ is called $d$- normal if its deficiency subspace is finite dimensional $( d ( A) = \mathop{\rm dim} {} ^ \perp R ( A) < + \infty )$. Operators that are either $n$- normal or $d$- normal are sometimes called semi-Fredholm operators. For an operator $A$ to be $n$- normal it is necessary and sufficient that the pre-image of every compact set in $R ( A)$ is locally compact.

Suppose that $X$ is compactly imbedded in a Banach space $X _ {0}$. For $A$ to be $n$- normal it is necessary and sufficient that there is an a priori estimate

$$\| x \| _ {X} \leq a \| x \| _ {X _ {0} } + b \| A x \| _ {Y} , \ x \in D ( A) .$$

It turns out that an operator $A$ is $n$- normal if and only if $A ^ {*}$ is $d$- normal. Then $n ( A) = d ( A ^ {*} )$. Consequently, if $X ^ {*}$ is compactly imbedded in a Banach space $Z$, then $A$ is $d$- normal if and only if there is an a priori estimate

$$\| f \| _ {Y ^ {*} } \leq a \| f \| _ {Z} + b \| A ^ {*} f \| _ {X ^ {*} } ,\ \ f \in D ( A ^ {*} ) .$$

The pair of numbers $( n ( A) , d ( A) )$ is called the $d$- characteristic of $A$. If a normally-solvable operator $A$ is $n$- normal or $d$- normal, the number

$$\chi ( a) = n ( A) - d ( A)$$

is called the index of the operator $A$. The properties of being $n$- normal and $d$- normal are stable: If $A$ is $n$- normal (or $d$- normal) and $B$ is a linear operator of small norm or completely continuous, then $A + B$ is also $n$- normal (respectively, $d$- normal).

References

[1]	F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))
[2]	F. Atkinson, "Normal solvability of equations in Banach space" Mat. Sb. , 28 : 1 (1951) pp. 3–14 (In Russian)
[3]	S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian)

References

[a1]	I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "The basic propositions on defect numbers, root numbers and indices of linear operators" Transl. Amer. Math. Soc. (2) , 13 (1960) pp. 185–264 Uspekhi Mat. Nauk , 12 (1957) pp. 43–118
[a2]	S. Goldberg, "Unbounded linear operators" , McGraw-Hill (1966)
[a3]	T. Kato, "Perturbation theory for nullity, deficiency and other quantities of linear operators" J. d'Anal. Math. , 6 (1958) pp. 261–322
[a4]	S.G. Krein, "Linear equations in Banach spaces" , Birkhäuser (1982) (Translated from Russian)