A strongly continuous semigroup (more precisely, strongly continuous operator semigroup, sometimes referred to as - semigroup ) is an object from the mathematical branch of functional analysis. Special cases of the strongly continuous semigroup are the norm continuous semigroup and the analytic semigroup.
A family of continuous linear mappings of a real or complex Banach space into itself, which the three properties
Met, ie strongly continuous semigroup. If we replace 3 by the stronger requirement
Is the name of the family norm continuous semigroup.
Can one continue the holomorphic semigroup on a sector, it is called analytic or holomorphic.
These semigroups play an important role in the ( abstract ) theory of evolution equations.
Be a continuous linear operator, then defining
The series converges absolutely and therefore defines a family of continuous linear operators. This family is a norm continuous semigroup and thus in particular also a strongly continuous semigroup.
Classification of strongly continuous semigroups
At any strongly continuous semigroup exist a and a, such that for all the assessment
Applies. Here denotes the operator norm on the Banach space of continuous linear endomorphisms of. We call the semigroup
- As a contraction semigroup, if it is satisfied for and,
- As limited semigroup if above inequality for one and applies
- As a quasi- contractive semigroup, if the above inequality for and satisfied.
The infimum over all possible, so called growth barrier.
If one considers instead, it is called strongly continuous groups.
Strongly continuous semigroups can be continued under certain circumstances to sectors in the complex plane. Such semigroups are called analytic.
Let be a strongly continuous semigroup. As infinitesimal generator or infinitesimal generator of some sort of illustration
Is a densely defined, closed, linear operator.
Is bounded if and only if even converges in the operator norm to the identity.
The abstract Cauchy problem
For the initial value and a continuously differentiable function is the function
For the spectrum of the generator applies: If, then applies, with the growth bound of the semigroup.
The resolvent of coincides with the Laplace transform of the semigroup, ie for and all.
Set of the Hille- Yosida
Of particular interest is whether a given operator of the infinitesimal generator is a strongly continuous semigroup. This question is answered completely by the set of Hille - Yosida:
A linear operator is exactly the infinitesimal generator of a strongly continuous semigroup which satisfies the estimate, if is complete and tightly defined subset of the resolvent set of is and
A use case is that you want to solve the evolution equation with a given differential operator. The set of Hille - Yosida states that one needs to investigate this, the resolvent equation, which then leads to elliptic problems. Can you solve the elliptic problem, it is easy to solve the evolution problem.
The theory of strongly continuous semigroups developed from the observation of the Cauchy problem. The simplest form of the Cauchy problem is the question of whether there exists a differentiable function for a given and an initial value, the
Met. From the theory of ordinary differential equations is obtained that is uniquely given by. This can now be generalized by considering the problem in higher dimensions, so as the initial value and as a matrix selected. Again, the solution is of
Here, the matrix exponential is defined by as in the real. The Cauchy problem can also be placed on a Banach space, is where and when an operator is selected on. Is a bounded operator, then again with the solution of the Cauchy problem. Occurring in the application of the Laplacian operator, such as the issue of a generalization to raise discontinuous operators, because in this case the sum is generally not converge. This results in the problem of how to define the exponential function in case of an unlimited operator. Independently Einar Hille and Kosaku Yosida could present a solution to the year 1948:
Approach of Hille: Starting from the current in the real identity is obtained. This representation has the advantage that the resolvent is bounded and thus appear on the right side only bounded operators. Hille was able to show that in certain circumstances the limit of this sequence exists. Consider a strongly continuous semigroup, as defined in the introduction, with their producer, it satisfies the equation.
Yosida approximation: Yosidas idea was to define the ( unbounded ) operator through a sequence of bounded operators. To this end, he sat and showed that in converges pointwise. Furthermore, produce as bounded operators with strongly continuous semigroups which converge pointwise for each in against an operator. The family of operators is in fact a strongly continuous semigroup, and every strongly continuous semigroup can be approximated by the Yosida approximation.