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).

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