Exponential of an Operator

In class we recently learned that for a linear operator and function one can define the operator (where is the identity transformation). We also recently learned about the exponential of a matrix. My question is that for a linear operator can the operator be defined? (For example, like how is defined for a matrix ) (I tried searching for information on it but all I found was information on the exponential of a matrix). Thanks.

Re: Exponential of an Operator

What is here and how is defined for a matrix A?

Re: Exponential of an Operator

here is the constant, (the one used for the exponential function .

And for a matrix , is defined as follows:

Re: Exponential of an Operator

Quote:

Originally Posted by

**Assassin0071** And for a matrix

,

is defined as follows:

Thanks.

Quote:

Originally Posted by

**Assassin0071** My question is that for a linear operator

can the operator

be defined? (For example, like how

is defined for a matrix

) (I tried searching for information on it but all I found was information on the exponential of a matrix).

Are you asking about operators on infinite-dimensional vector spaces? Because on a finite-dimensional space, every linear operator is given by a matrix, so the exponentiation of an operator is determined by the exponentiation of its matrix.