Bounded operator

A mapping $ A $ of a topological vector space $ X $ into a topological vector space $ Y $ such that $ A (M) $ is a bounded subset in $ Y $ for any bounded subset $ M $ of $ X $. Every operator $ A : X \rightarrow Y $, continuous on $ X $, is a bounded operator. If $ A : X \rightarrow Y $ is a linear operator, then for $ A $ to be bounded it is sufficient that there exists a neighbourhood $ U \subset X $ such that $ A (U) $ is bounded in $ Y $. Suppose that $ X $ and $ Y $ are normed linear spaces and that the linear operator $ A : X \rightarrow Y $ is bounded. Then

$$ \gamma = \ \sup _ {\| x \| \leq 1 } \| A x \| < \infty . $$

This number is called the norm of the operator $ A $ and is denoted by $ \| A \| $. Then

$$ \| A x \| \leq \| A \| \cdot \| x \| , $$

and $ \| A \| $ is the smallest constant $ C $ such that

$$ \| A x \| \leq C \| x \| $$

for any $ x \in X $. Conversely, if this inequality is satisfied, then $ A $ is bounded. For linear operators mapping a normed space $ X $ into a normed space $ Y $, the concepts of boundedness and continuity are equivalent. This is not the case for arbitrary topological vector spaces $ X $ and $ Y $, but if $ X $ is bornological and $ Y $ is a locally convex space, then the boundedness of a linear operator $ A : X \rightarrow Y $ implies its continuity. If $ H $ is a Hilbert space and $ A : H \rightarrow H $ is a bounded symmetric operator, then the quadratic form $ \langle A x , x \rangle $ is bounded on the unit ball $ K _ {1} = \{ {x } : {\| x \| \leq 1 } \} $. The numbers

$$ \beta = \ \sup _ {\| x \| \leq 1 } \langle A x , x \rangle \ \ \textrm{ and } \ \ \alpha = \inf _ {\| x \| \leq 1 } \langle A x , x \rangle $$

are called the upper and lower bounds of the operator $ A $. The points $ \alpha $ and $ \beta $ belongs to the spectrum of $ A $, and the whole spectrum lies in the interval $ [ \alpha , \beta ] $. Examples of bounded operators are: the projection operator (projector) onto a complemented subspace of a Banach space, and an isometric operator acting on a Hilbert space.

If the space $ X $ and $ Y $ have the structure of a partially ordered set, for example are vector lattices (cf. Vector lattice), then a concept of order-boundedness of an operator can be introduced, besides the topological boundedness considered above. An operator $ A : X \rightarrow Y $ is called order-bounded if $ A (M) $ is an order-bounded set in $ Y $ for any order-bounded set $ M $ in $ X $. Examples: an isotone operator, i.e. an operator such that $ x \leq y $ implies $ A x \leq A y $.

References

[1]	L.A. Lyusternik, V.I. Sobolev, "Elemente der Funktionalanalysis" , Akademie Verlag (1955) (Translated from Russian)
[2]	W. Rudin, "Functional analysis" , McGraw-Hill (1973)
[3]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1967)

References

[a1]	A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980)
[a2]	A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)