Operator on a set spaned the space

[LikeMath]

Hi there,

Let X be a Hilbert (Banach) space, and spanned by a set S, say.
Let A be linear bounded operator on X into itself.
Suppose that the operator is well known on S, that is
Aa_i=b_i for all a_i\in S.
First, is this operator unique on X? if yes, can we find Aa, for general element a in X, in terms of b_i.

Thanks in advance

 

[micromass]

spanned by a set S

What do you mean with this?? I ask because you might mean something completely different that I am thinking right now.

 

[LikeMath]

That is, the space is generated by the closure of all linear combinations from S.

 

[micromass]

Ah, I suspected something like that.

The answer is yes, the operator will be unique.

You can see that is two stages: suppose that T is unique on S, then it's also unique on linear combinations of S. Indeed, take a linear combination a=\sum \lambda_i a_i, then we have

T(a)=\sum \lambda_i T(a_i)

Then use that if T is unique on a dense subset, then it's unique on the entire set. Indeed, take x in the closure, then there is a sequence x_n\rightarrow x. It must hold that

T(x_n)\rightarrow T(x)

 

[LikeMath]

Thank you.
I completely agree with you. But what about the second part of the question, that is
can we find the operator on the whole space X?