Circles in the spectrum and numerical ranges
Vladimir Müller
Ljubljana, 2017
Vladimir Müller
Circles in the spectrum and numerical ranges
joint work with Yu. Tomilov, IM PAN, Warsaw
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Arveson 1966) Let U ∈ B(H) be a unitary operator. The
following statements are equivalent:
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Arveson 1966) Let U ∈ B(H) be a unitary operator. The
following statements are equivalent:
(i) σ(U) = T;
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Arveson 1966) Let U ∈ B(H) be a unitary operator. The
following statements are equivalent:
(i) σ(U) = T;
(ii) for every n ∈ N there exists a unit vector x ∈ H such that
x, Ux, . . . , U n x are orthogonal.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Hamdan 2013) Let µ be a Rajchman measure.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Hamdan 2013) Let µ be a Rajchman measure. The following
statements are equivalent:
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Hamdan 2013) Let µ be a Rajchman measure. The following
statements are equivalent:
(i) supp µ = T;
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Hamdan 2013) Let µ be a Rajchman measure. The following
statements are equivalent:
(i) supp µ = T;
(ii) for every ε > 0 there exists a function f ∈ L1 (µ), kf k1 = 1,
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Hamdan 2013) Let µ be a Rajchman measure. The following
statements are equivalent:
(i) supp µ = T;
(ii) for every ε > 0 there exists a function f ∈ L1 (µ), kf k1 = 1,
f ≥0
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Hamdan 2013) Let µ be a Rajchman measure. The following
statements are equivalent:
(i) supp µ = T;
(ii) for every ε > 0 there exists a function f ∈ L1 (µ), kf k1 = 1,
f ≥ 0 such that supn |f̂ (n)| < ε
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Hamdan 2013) Let µ be a Rajchman measure. The following
statements are equivalent:
(i) supp µ = T;
(ii) for every ε > 0 there exists a function f ∈ L1 (µ), kf k1 = 1,
f ≥ 0 such that supn |f̂ (n)| < ε
Theorem
Let U ∈ B(H) be a unitary operator such that T n → 0
The following statements are equivalent:
Vladimir Müller
(WOT ).
Circles in the spectrum and numerical ranges
Theorem
(Hamdan 2013) Let µ be a Rajchman measure. The following
statements are equivalent:
(i) supp µ = T;
(ii) for every ε > 0 there exists a function f ∈ L1 (µ), kf k1 = 1,
f ≥ 0 such that supn |f̂ (n)| < ε
Theorem
Let U ∈ B(H) be a unitary operator such that T n → 0
The following statements are equivalent:
(i) σ(U) = T;
Vladimir Müller
(WOT ).
Circles in the spectrum and numerical ranges
Theorem
(Hamdan 2013) Let µ be a Rajchman measure. The following
statements are equivalent:
(i) supp µ = T;
(ii) for every ε > 0 there exists a function f ∈ L1 (µ), kf k1 = 1,
f ≥ 0 such that supn |f̂ (n)| < ε
Theorem
Let U ∈ B(H) be a unitary operator such that T n → 0 (WOT ).
The following statements are equivalent:
(i) σ(U) = T;
(ii) for every ε > 0 there exists a unit vector x ∈ H such that
supn≥1 |hU n x, xi| < ε.
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T1 , . . . , Tn ∈ B(H).
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T1 , . . . , Tn ∈ B(H). The joint numerical range is defined by
W (T1 , . . . , Tn ) = (hT1 x, xi, . . . , hTn x, xi) : x ∈ H, kxk = 1
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T1 , . . . , Tn ∈ B(H). The joint numerical range is defined by
W (T1 , . . . , Tn ) = (hT1 x, xi, . . . , hTn x, xi) : x ∈ H, kxk = 1
We write shortly T = (T1 , . . . , Tn ) ∈ B(H)n .
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T1 , . . . , Tn ∈ B(H). The joint numerical range is defined by
W (T1 , . . . , Tn ) = (hT1 x, xi, . . . , hTn x, xi) : x ∈ H, kxk = 1
We write shortly T = (T1 , . . . , Tn ) ∈ B(H)n . For x, y ∈ H write
hT x, y i = (hT1 x, y i, . . . , hTn x, y i) ∈ Cn
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T1 , . . . , Tn ∈ B(H). The joint numerical range is defined by
W (T1 , . . . , Tn ) = (hT1 x, xi, . . . , hTn x, xi) : x ∈ H, kxk = 1
We write shortly T = (T1 , . . . , Tn ) ∈ B(H)n . For x, y ∈ H write
hT x, y i = (hT1 x, y i, . . . , hTn x, y i) ∈ Cn
Definition
Let T = (T1 , . . . , Tn ) ∈ B(H)n . The essential numerical range is
defined by
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T1 , . . . , Tn ∈ B(H). The joint numerical range is defined by
W (T1 , . . . , Tn ) = (hT1 x, xi, . . . , hTn x, xi) : x ∈ H, kxk = 1
We write shortly T = (T1 , . . . , Tn ) ∈ B(H)n . For x, y ∈ H write
hT x, y i = (hT1 x, y i, . . . , hTn x, y i) ∈ Cn
Definition
Let T = (T1 , . . . , Tn ) ∈ B(H)n . The essential numerical range is
defined by
We (T ) = λ ∈ Cn : there exists an orthonomal sequence
(xk ) ⊂ H such that λ = lim hT xk , xk i
k →∞
Vladimir Müller
Circles in the spectrum and numerical ranges
If n = 1 then: W (T ) is convex
Vladimir Müller
Circles in the spectrum and numerical ranges
If n = 1 then: W (T ) is convex
conv σ(T ) ⊂ W (T )
Vladimir Müller
Circles in the spectrum and numerical ranges
If n = 1 then: W (T ) is convex
conv σ(T ) ⊂ W (T )
Int W (T ) ⊂ W (T )
Vladimir Müller
Circles in the spectrum and numerical ranges
If n = 1 then: W (T ) is convex
conv σ(T ) ⊂ W (T )
Int W (T ) ⊂ W (T )
Int conv σ(T ) ⊂ W (T )
Vladimir Müller
Circles in the spectrum and numerical ranges
If n = 1 then: W (T ) is convex
conv σ(T ) ⊂ W (T )
Int W (T ) ⊂ W (T )
Int conv σ(T ) ⊂ W (T )
If n ≥ 2 then in general W (T1 , . . . , Tn ) is not convex
Vladimir Müller
Circles in the spectrum and numerical ranges
If n = 1 then: W (T ) is convex
conv σ(T ) ⊂ W (T )
Int W (T ) ⊂ W (T )
Int conv σ(T ) ⊂ W (T )
If n ≥ 2 then in general W (T1 , . . . , Tn ) is not convex
We (T1 , . . . , Tn ) is convex (Li, Poon 2009)
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Wrobel 1988) T = (T1 , . . . , Tn ) ∈ B(H)n commuting operators
then
conv σ(T1 , . . . , Tn ) ⊂ W (T1 , . . . , Tn )
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T = (T1 , . . . , Tn ) ∈ B(H)n .
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Then
Int (We (T )) ⊂ W (T ).
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Then
Int (We (T )) ⊂ W (T ).
Moreover, if µ ∈ Int (We (T ))
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Then
Int (We (T )) ⊂ W (T ).
Moreover, if µ ∈ Int (We (T )) then for every subspace M ⊂ H of
a finite codimension there exists x ∈ M such that kxk = 1 and
hT1 x, xi, . . . , hTn x, xi = µ.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Then
Int (We (T )) ⊂ W (T ).
Moreover, if µ ∈ Int (We (T )) then for every subspace M ⊂ H of
a finite codimension there exists x ∈ M such that kxk = 1 and
hT1 x, xi, . . . , hTn x, xi = µ.
Corollary
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Let µ ∈ Int (We (T )).
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Then
Int (We (T )) ⊂ W (T ).
Moreover, if µ ∈ Int (We (T )) then for every subspace M ⊂ H of
a finite codimension there exists x ∈ M such that kxk = 1 and
hT1 x, xi, . . . , hTn x, xi = µ.
Corollary
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Let µ ∈ Int (We (T )). Then there
exists an infinite-dimensional subspace L ⊂ H
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Then
Int (We (T )) ⊂ W (T ).
Moreover, if µ ∈ Int (We (T )) then for every subspace M ⊂ H of
a finite codimension there exists x ∈ M such that kxk = 1 and
hT1 x, xi, . . . , hTn x, xi = µ.
Corollary
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Let µ ∈ Int (We (T )). Then there
exists an infinite-dimensional subspace L ⊂ H such that
PL Tj PL = µj PL
Vladimir Müller
(j = 1, . . . , n).
Circles in the spectrum and numerical ranges
Theorem
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Then
Int (We (T )) ⊂ W (T ).
Moreover, if µ ∈ Int (We (T )) then for every subspace M ⊂ H of
a finite codimension there exists x ∈ M such that kxk = 1 and
hT1 x, xi, . . . , hTn x, xi = µ.
Corollary
Let T = (T1 , . . . , Tn ) ∈ B(H)n . Let µ ∈ Int (We (T )). Then there
exists an infinite-dimensional subspace L ⊂ H such that
PL Tj PL = µj PL
(j = 1, . . . , n).
Equivalently, Int (We (T )) ⊂ W∞ (T ).
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T ∈ B(H)n . Then W∞ (T ) is the set of all λ ∈ Cn for which
there exists an infinite-dimensional subspace L ⊂ H such that
PL T PL = λPL
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T ∈ B(H)n . Then W∞ (T ) is the set of all λ ∈ Cn for which
there exists an infinite-dimensional subspace L ⊂ H such that
PL T PL = λPL
W∞ (T ) is always convex
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T ∈ B(H)n . Then W∞ (T ) is the set of all λ ∈ Cn for which
there exists an infinite-dimensional subspace L ⊂ H such that
PL T PL = λPL
W∞ (T ) is always convex
it may be empty
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T ∈ B(H)n . Then W∞ (T ) is the set of all λ ∈ Cn for which
there exists an infinite-dimensional subspace L ⊂ H such that
PL T PL = λPL
W∞ (T ) is always convex
it may be empty
if W∞ (T ) = ∅ then the n-tuple T is "degenerated"
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T ∈ B(H)n . Then W∞ (T ) is the set of all λ ∈ Cn for which
there exists an infinite-dimensional subspace L ⊂ H such that
PL T PL = λPL
W∞ (T ) is always convex
it may be empty
if W∞ (T ) = ∅ then the n-tuple T is "degenerated"
Theorem
Let T ∈ B(H)n . Then
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T ∈ B(H)n . Then W∞ (T ) is the set of all λ ∈ Cn for which
there exists an infinite-dimensional subspace L ⊂ H such that
PL T PL = λPL
W∞ (T ) is always convex
it may be empty
if W∞ (T ) = ∅ then the n-tuple T is "degenerated"
Theorem
Let T ∈ B(H)n . Then
S
(i) We (T ) = K∈K(H)n W∞ (T + K)
Vladimir Müller
Circles in the spectrum and numerical ranges
Definition
Let T ∈ B(H)n . Then W∞ (T ) is the set of all λ ∈ Cn for which
there exists an infinite-dimensional subspace L ⊂ H such that
PL T PL = λPL
W∞ (T ) is always convex
it may be empty
if W∞ (T ) = ∅ then the n-tuple T is "degenerated"
Theorem
Let T ∈ B(H)n . Then
S
(i) We (T ) = K∈K(H)n W∞ (T + K)
(ii) there exists an n-tuple K of compact operators such that
We (T ) = W∞ (T + K)
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H)n . Then
Int conv We (T ) ∪ σp (T ) ⊂ W (T )
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H)n . Then
Int conv We (T ) ∪ σp (T ) ⊂ W (T )
Corollary
Let T = (T1 , . . . , Tn ) ∈ B(H)n be a commuting tuple. Then
Int conv σ(T ) ⊂ W (T )
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) and let λ ∈ Int σ̂(T ).
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) and let λ ∈ Int σ̂(T ). Then
(λ, λ2 , . . . , λn ) ∈ Int (We (T , T 2 , . . . , T n )).
for all n ∈ N.
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T ∈ B(H), 0 ∈ Int σ̂(T ).
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T ∈ B(H), 0 ∈ Int σ̂(T ). Then for every n ∈ N there exists a
unit vector x ∈ H such that
x ⊥ Tx, . . . , T n x
Corollary
Let T ∈ B(H), 0 ∈ Int σ̂(T ).
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T ∈ B(H), 0 ∈ Int σ̂(T ). Then for every n ∈ N there exists a
unit vector x ∈ H such that
x ⊥ Tx, . . . , T n x
Corollary
Let T ∈ B(H), 0 ∈ Int σ̂(T ). Then for every n ∈ N there exists
an infinite-dimensional subspace L ⊂ H
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T ∈ B(H), 0 ∈ Int σ̂(T ). Then for every n ∈ N there exists a
unit vector x ∈ H such that
x ⊥ Tx, . . . , T n x
Corollary
Let T ∈ B(H), 0 ∈ Int σ̂(T ). Then for every n ∈ N there exists
an infinite-dimensional subspace L ⊂ H such that PL T j PL = 0
for j = 1, . . . , n.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
(i) T ⊂ σπe (T );
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
(i) T ⊂ σπe (T );
(ii) for all ε > 0 and n ∈ N there exists x ∈ H such that
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
(i) T ⊂ σπe (T );
(ii) for all ε > 0 and n ∈ N there exists x ∈ H such that
|hT m x, T j xi| < ε,
Vladimir Müller
1 ≤ m, j ≤ n − 1, m 6= j,
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
(i) T ⊂ σπe (T );
(ii) for all ε > 0 and n ∈ N there exists x ∈ H such that
|hT m x, T j xi| < ε,
and
1
2
≤ kT j xk ≤ 2,
1 ≤ m, j ≤ n − 1, m 6= j,
0 ≤ j ≤ n − 1;
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
(i) T ⊂ σπe (T );
(ii) for all ε > 0 and n ∈ N there exists x ∈ H such that
|hT m x, T j xi| < ε,
1 ≤ m, j ≤ n − 1, m 6= j,
and 12 ≤ kT j xk ≤ 2,
0 ≤ j ≤ n − 1;
(iii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
(i) T ⊂ σπe (T );
(ii) for all ε > 0 and n ∈ N there exists x ∈ H such that
|hT m x, T j xi| < ε,
1 ≤ m, j ≤ n − 1, m 6= j,
and 12 ≤ kT j xk ≤ 2,
0 ≤ j ≤ n − 1;
(iii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
x ⊥ Tx, T 2 x, . . . , T n−1 x,
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
(i) T ⊂ σπe (T );
(ii) for all ε > 0 and n ∈ N there exists x ∈ H such that
|hT m x, T j xi| < ε,
1 ≤ m, j ≤ n − 1, m 6= j,
and 12 ≤ kT j xk ≤ 2,
0 ≤ j ≤ n − 1;
(iii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
x ⊥ Tx, T 2 x, . . . , T n−1 x,
|hT m x, T j xi| < ε,
Vladimir Müller
1 ≤ m, j ≤ n − 1, m 6= j,
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
(i) T ⊂ σπe (T );
(ii) for all ε > 0 and n ∈ N there exists x ∈ H such that
|hT m x, T j xi| < ε,
1 ≤ m, j ≤ n − 1, m 6= j,
and 12 ≤ kT j xk ≤ 2,
0 ≤ j ≤ n − 1;
(iii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
x ⊥ Tx, T 2 x, . . . , T n−1 x,
|hT m x, T j xi| < ε,
1 ≤ m, j ≤ n − 1, m 6= j,
1 − ε < kT j xk < 1 + ε,
Vladimir Müller
0 ≤ j ≤ n − 1,
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H). The following statements are equivalent.
(i) T ⊂ σπe (T );
(ii) for all ε > 0 and n ∈ N there exists x ∈ H such that
|hT m x, T j xi| < ε,
1 ≤ m, j ≤ n − 1, m 6= j,
and 12 ≤ kT j xk ≤ 2,
0 ≤ j ≤ n − 1;
(iii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
x ⊥ Tx, T 2 x, . . . , T n−1 x,
|hT m x, T j xi| < ε,
1 ≤ m, j ≤ n − 1, m 6= j,
1 − ε < kT j xk < 1 + ε,
0 ≤ j ≤ n − 1,
and kT n x − xk < ε.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) satisfy r (T ) ≤ 1. The following statements are
equivalent:
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) satisfy r (T ) ≤ 1. The following statements are
equivalent:
(i) T ⊂ σ(T ).
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) satisfy r (T ) ≤ 1. The following statements are
equivalent:
(i) T ⊂ σ(T ).
(ii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
|hT m x, T j xi| < ε,
Vladimir Müller
1 ≤ m, j ≤ n − 1, m 6= j,
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) satisfy r (T ) ≤ 1. The following statements are
equivalent:
(i) T ⊂ σ(T ).
(ii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
|hT m x, T j xi| < ε,
and
1
2
≤ kT j xk ≤ 2,
1 ≤ m, j ≤ n − 1, m 6= j,
0 ≤ j ≤ n − 1;
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) satisfy r (T ) ≤ 1. The following statements are
equivalent:
(i) T ⊂ σ(T ).
(ii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
|hT m x, T j xi| < ε,
and
1
2
≤ kT j xk ≤ 2,
1 ≤ m, j ≤ n − 1, m 6= j,
0 ≤ j ≤ n − 1;
(iii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) satisfy r (T ) ≤ 1. The following statements are
equivalent:
(i) T ⊂ σ(T ).
(ii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
|hT m x, T j xi| < ε,
and
1
2
≤ kT j xk ≤ 2,
1 ≤ m, j ≤ n − 1, m 6= j,
0 ≤ j ≤ n − 1;
(iii) for all ε > 0 and all n ∈ N there exists a unit vector x ∈ H
such that
x ⊥ Tx, T 2 x, . . . , T n−1 x,
|hT m x, T j xi| < ε,
j
1 − ε < kT xk < 1 + ε,
1 ≤ m, j ≤ n − 1, m 6= j,
1 ≤ j ≤ n − 1,
and kT n x − xk < ε.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) and let T n → 0 in the weak operator topology.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) and let T n → 0 in the weak operator topology.
Suppose that (0, . . . , 0) ∈ We (T , . . . , T n ) for all n ∈ N.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) and let T n → 0 in the weak operator topology.
Suppose that (0, . . . , 0) ∈ We (T , . . . , T n ) for all n ∈ N. Then for
every ε > 0 there exists an infinite-dimensional subspace L of
H
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H) and let T n → 0 in the weak operator topology.
Suppose that (0, . . . , 0) ∈ We (T , . . . , T n ) for all n ∈ N. Then for
every ε > 0 there exists an infinite-dimensional subspace L of
H such that
sup kPL T n PL k ≤ ε
and
n≥1
Vladimir Müller
lim kPL T n PL k = 0.
n→∞
Circles in the spectrum and numerical ranges
Corollary
Let T ∈ B(H), and let T n → 0 in the weak operator topology.
Suppose that 0 ∈ Int σ̂(T ).
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T ∈ B(H), and let T n → 0 in the weak operator topology.
Suppose that 0 ∈ Int σ̂(T ). Then for every ε > 0 there exists an
infinite-dimensional subspace L of H
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T ∈ B(H), and let T n → 0 in the weak operator topology.
Suppose that 0 ∈ Int σ̂(T ). Then for every ε > 0 there exists an
infinite-dimensional subspace L of Hsuch that
sup kPL T n PL k ≤ ε
and
n≥1
Vladimir Müller
lim kPL T n PL k = 0.
n→∞
Circles in the spectrum and numerical ranges
Corollary
Let T be a unitary operator on H such that T n → 0 in the weak
operator topology.
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T be a unitary operator on H such that T n → 0 in the weak
operator topology. Then the following conditions are equivalent.
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T be a unitary operator on H such that T n → 0 in the weak
operator topology. Then the following conditions are equivalent.
(i) σ(T ) = T.
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T be a unitary operator on H such that T n → 0 in the weak
operator topology. Then the following conditions are equivalent.
(i) σ(T ) = T.
(ii) for every ε > 0 there exists x ∈ H, kxk = 1, with
sup |hT n x, xi| < ε.
n≥1
Vladimir Müller
Circles in the spectrum and numerical ranges
Corollary
Let T be a unitary operator on H such that T n → 0 in the weak
operator topology. Then the following conditions are equivalent.
(i) σ(T ) = T.
(ii) for every ε > 0 there exists x ∈ H, kxk = 1, with
sup |hT n x, xi| < ε.
n≥1
(iii) for every ε > 0 there exists an infinite-dimensional
subspace L ⊂ H such that
sup kPL T n PL k ≤ ε
and
n≥1
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lim kPL T n PL k = 0.
n→∞
Circles in the spectrum and numerical ranges
Theorem
(Bourin 2003) Let T ∈ B(H). Suppose that We (T ) ⊃ D. Then
for every strict contraction C on a separable Hilbert space (i.e.,
kCk < 1)
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
(Bourin 2003) Let T ∈ B(H). Suppose that We (T ) ⊃ D. Then
for every strict contraction C on a separable Hilbert space (i.e.,
kCk < 1), there exists a subspace L ⊂ H such that the
compression TL is unitarily equivalent to C.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H), D ⊂ σ̂(T ), let n ∈ N. Then for every strict
contraction C 0
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H), D ⊂ σ̂(T ), let n ∈ N. Then for every strict
contraction C 0 there exists a subspace L ⊂ H and C ∈ B(L)
unitarily equivalent to C 0 such that
(T j )L = C j
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H), D ⊂ σ̂(T ), let n ∈ N. Then for every strict
contraction C 0 there exists a subspace L ⊂ H and C ∈ B(L)
unitarily equivalent to C 0 such that
(T j )L = C j
for j = 1, . . . , n.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H), T n → 0 (WOT ), σ(T ) ⊃ T, let ε > 0.
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H), T n → 0 (WOT ), σ(T ) ⊃ T, let ε > 0. Then for
every strict contraction C 0 there exists a subspace L ⊂ H and
C ∈ B(L) unitarily equivalent to C 0
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H), T n → 0 (WOT ), σ(T ) ⊃ T, let ε > 0. Then for
every strict contraction C 0 there exists a subspace L ⊂ H and
C ∈ B(L) unitarily equivalent to C 0 such that
sup k(T j )L − C j k < ε
j
Vladimir Müller
Circles in the spectrum and numerical ranges
Theorem
Let T ∈ B(H), T n → 0 (WOT ), σ(T ) ⊃ T, let ε > 0. Then for
every strict contraction C 0 there exists a subspace L ⊂ H and
C ∈ B(L) unitarily equivalent to C 0 such that
sup k(T j )L − C j k < ε
j
and
lim k(T j )L − C j k = 0
j→∞
Vladimir Müller
Circles in the spectrum and numerical ranges
THANK YOU FOR YOUR ATTENTION
Vladimir Müller
Circles in the spectrum and numerical ranges