SOME REMARKS ON SYMMETRIC UNBOUNDED OPERATORS
RAUL QUIROGA-BARRANCO
In what follows H denotes a Hilbert space and T a linear operator on H defined
in a subspace D(T ). Any further assumptions on T , including its domain, are
explicitly stated in each case.
1. Preliminaries
A value λ ∈ C is called a regular value for T when the operator T − λI has a
bounded inverse defined on R(T − λI). The set of regular points of T is denoted
by π(T ). It is known (see [2]) that π(T ) is an open subset of C. For every λ ∈ π(T )
we define the deficiency space for T at λ by R(T − λI)⊥ and the dimension of such
space is called the deficiency index of T at λ and it is denoted by dλ (T ).
Proposition 1.1. Let T be a linear operator on H. If for some λ ∈ π(T ) we have
dλ (T ) = 0, then T is closed.
Proof. By hypothesis, we have a bounded operator (T − λI)−1 : H → H which is
necessarily closed since it is defined in all of H. Hence, T − λI is closed and so T
is closed as well.
A value λ ∈ C is said to belong to the resolvent set ρ(T ) of T when the operator
has a bounded inverse defined on R(T − λI) = H. The set σ(T ) = C \ ρ(T ) is called
the spectrum of T .
Proposition 1.2. Let T be a linear operator on H. If T is not closed, then we
have ρ(T ) = ∅. If T is closed and λ ∈ π(T ), then dλ (T ) = 0 if and only if λ ∈ ρ(T ).
Proof. The first part is a consequence of Proposition 1.1. The second claim follows
from the definition of the resolvent set.
Because of the previous result the spectral is trivial for non-closed operators.
Hence, the spectrum and resolvent set is considered only for closed operators.
The operator T is called symmetric when it satisfies
hT x, yi = hx, T yi
for every x, y ∈ D(T ).
Proposition 1.3 ([2, Proposition 3.2]). Let T be a symmetric operator on H. Then
(1) C \ R ⊂ π(T ).
(2) If T − mI ≥ 0 for some m ∈ R, then (−∞, m) ⊂ π(T ).
(3) If T is densely defined and λ ∈ π(T ), then R(T ∗ − λI) = H.
Because of the following result one can simplify the use of the deficiency indices.
Proposition 1.4 ([2, Proposition 2.4]). Let T be a closable linear operator on H.
Then the map λ 7→ dλ (T ) is constant on the connected components of π(T ).
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RAUL QUIROGA-BARRANCO
2. To further explain
Let T be a symmetric linear operator on H. Hence, ±i ∈ π(T ) and so T − iI and
T + iI admit bounded inverses defined on R(T − iI) and R(T + iI), respectively.
The Cayley transformation U = UT is the map
U = (T − iI)(t + iI)−1 : R(T + iI) → R(T − iI)
which is known to be an isometry. For this map to be extendable to an isometry
H → H we need two conditions
• both R(T − iI) and R(T + iI) to be closed, and
• di (T ) = d−i (T ).
The first condition does not hold in general for arbitrary T . But we have the
following facts.
• If T is symmetric and densely defined, then T is closable.
• If T is closed and symmetric, then R(T − λI) is closed for every λ ∈ π(T ).
This allows to extend a symmetric densely defined operator to a self-adjoint one
using the Cayley transform when di (T ) = d−i (T ). But the argument does require
T to be at least closed, e.g. by requiring that T is densely defined.
References
[1] W. Rudin, Functional Analysis.
[2] K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space.