EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 7, No. 2, 2014, 129-130
ISSN 1307-5543 – www.ejpam.com

The Distance From a Point to a Compact Convex Set

M. T. Heydari

Department of Mathematics, College of Sciences, Yasouj University, Yasouj, 75914, Iran

Abstract. Let K be a compact convex subset of the plane and λ ∈ C\K , then

dist(λ, K) = ‖(λ− Nµ)
−1‖−1,

where µ is the Lebesgue measure concentrated on K and Nµ be the multiplication operator on L2(µ).

2010 Mathematics Subject Classifications: 47A12, 11F23

Key Words and Phrases: compact convex set, distance, Numerical range

1. Main Result

Let K be a compact convex subset of the plane and µ be the Lebesgue measure concentrated
on K , i.e., µ= m2|K . Define Nµ on L2(µ) by Nµ f = z f for each f in L2(µ). It is easy to check
that Nµ is normal. Let s ∈ K and put Un = B(s, 1

n), the disc with center at s and radius 1
n , so

µ(Un) 6= 0. Since µ is regular then µ(Un)<∞. Now define

fn =
1
p

µ(Un)
χUn

,

so ‖ fn‖2 = 1 and ‖(Nµ − s) fn‖2 −→ 0, that is s ∈ σ(Nµ). Let s ∈ K c , then there is an open set
U with µ(U) = 0 and s ∈ U . Define

ψ(z) =

¨

(s− z)−1 if z ∈ U c;

0 if z ∈ U .

There is r > 0 such that B(s, r) ⊂ U . If z ∈ U c then 1
|s−z| <

1
r . Therefore ‖ψ‖∞ ≤

1
r a.e. and

so ψ ∈ L∞(µ). Define the operator T on L2(µ) by T ( f ) =ψ f , then we have

(s− Nµ)T = T (s− Nµ) = I

Email address: heydari@yu.ac.ir

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REFERENCES 130

a.e.. Thus s is not in σ(Nµ) and so σ(Nµ) = K . Thus for λ ∈ C\K we have (see [1, Proposition
3.9 p.198]):

‖(λ− Nµ)
−1‖−1 ≤ dist(λ, K). (1)

To prove the inverse inequality, we need to the following concepts which can be found in [2].
For a bounded linear operator T on a Hilbert space H , the numerical range W (T ) is the

image of the unit sphere of H under the quadratic form x →< T x , x > associated with the
operator. More precisely,

W (T ) = {< T x , x >: x ∈H ,‖x‖= 1}

Thus the numerical range of an operator, like the spectrum, is a subset of the complex
plane whose geometrical properties should say something about the operator.

One of the most fundamental properties of the numerical range is its convexity, stated by
the famous Toeplitz-Hausdorff Theorem. Other important property of W (T ) is that its closure
contains the spectrum of the operator. W (T ) is a connected set and for normal operator N ,

W (N) = co(σ(N)), (2)

where σ(N) is the spectrum of N .
Also we need to the following Theorem which can be found in [3].

Theorem 1. Let T be a bounded linear operator T on a Hilbert space H and λ outside W (T ).
Then

dist(λ, W (T ))≤ ‖(λ− T )−1‖−1. (3)

For the operator Nµ as defined in the first paragraph, we have W (Nµ) = K and the above
Theorem implies that:

dist(λ, K)≤ ‖(λ− Nµ)
−1‖−1. (4)

Now the result follows from (1) and (4).

References

[1] J. B. Conway, A course in Functional Analysis, Second ed., Springer-Verlag, New York, 1985.

[2] P. R. Halmos. A Hilbert space problem Book, second edition, Springer, New York, 1982.

[3] F. J. Narcowich. Analytic properties of the boundary of the numerical range, Indiana Univer-
sity Mathematics Journal, 29, 67-77. 1980.