0] induces a

bounded operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on [L.

Let A be a fixed linear

bounded operator acting on the Banach space [C.

ran(C)] is a

bounded operator from ran(C) onto ran(C) with bounded inverse.

k]}, for any

bounded operator W : H [right arrow] H.

It is well known that A is a

bounded operator on the Banach space m of bounded sequences if and only if

Notice that, in the case that A is a

bounded operator, the left-definite theory is trivial but, when A is unbounded, the theory has substance.

Moreover, a look at the proof of Observation 2 reveals that every translation invariant, linear, and

bounded operator T : W [right arrow] [L.

m] is a

bounded operator and the following inequality holds:

0] induces a

bounded operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for every f [member of] [L.

1]) is a

bounded operator (and gives a norm estimate depending on the unknown constant K').

t] be a

bounded operator, and take [lambda] [member of] C.

Clearly for two Bessel sequences it is well-defined as linear

bounded operator because