Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.1
——–Big Basesand large diagonal operators
Big Bases May 2008
Ben MathesColby College
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.2
Big Bases and Large Diagonal Operators
2666666666666666666666666666664
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 13 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 14 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18
3777777777777777777777777777775
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.3
Overview
1 KalischOne dimensional example: Mx − VTwo dimensional example: Mx − V + i(Ny −W )
2 Sarason - WatermanInvariant subspaces of Mx + VInvariant subspaces of Mx − VSpectral synthesis!
3 Strictly cyclic algebrasSarason’s algebraTensor productsNew Examples
4 Substrictly cyclic algebrasSarason, Erdos, ...Idealsa substrict algebra
5 End
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.4
• Theorem
(Kalisch) Given any compact subset of the plane, there existsan operator whose spectrum equals that compact set andconsists entirely of simple point spectrum.
• We say that α is in the point spectrum of T when
Tv = αv
for some v 6= 0, and it is simple point spectra if thecorresponding eigenspace is one dimensional.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.5
First Kalisch paper......Mx − V
1
1t
A big basis......
{χ[t,1] : t ∈ [o,1)
}
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.6
apply Mx (red) and −V (blue), then add ......
1
1t
A continuum of eigenvectors for Mx − V ......
χ[t,1] 7→ tχ[t,1]
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.7
T = Mx − V
Theorem
(Kalisch) Take any closed subset E of (0,1), and letMEdenote the closed linear span of the correspondingeigenvectors. Then the restriction of T to this invariantsubspace has spectrum E and consists entirely of simple pointspectra.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.8
T = Mx − V + i(Ny −W )
• To accommodate sets with planar interior, move to L2(I)with I the unit square.
• Use the operator T = Mx − V + i(Ny −W ) whosespectrum is the closed unit square.
• Show that every α in the interior of I is simple pointspectra.
• Theorem
(Kalisch) Take any closed subset E contained in the interior ofI, and letME denote the closed linear span of thecorresponding eigenvectors. Then the restriction of T to thisinvariant subspace has spectrum E and consists entirely ofsimple point spectra.
• Technique of proof: here’s an operator, let’s roll up oursleeves and compute the spectrum!
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.9
Sarason and T = Mx + V
• Use V to map L2[0,1] bijectively onto the set A ofabsolutely continuous functions that vanish at the origin.
• Put a norm on A so that V becomes a unitary.• Observe that Mx + V is then unitarily equivalent to
multiplication by x on A• Since A is an algebra, find the closed ideals to
characterize the invariant subspaces.• Technique of proof: Banach algebras - function spaces
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.10
The relation of Sarason’s operator to Kalisch......
1
1t
Eigenvectors for T ∗......
{χ[0,t] : t ∈ (0,1]
}χ[0,t] 7→ tχ[0,t]
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.11
Like Kalisch, Waterman works with T = Mx − V(Waterman was a student of Kalisch)
• Characterize the algebra generated by T , the algebra of“large diagonal operators"
• The mappingχ[t,1] 7→ h(t)χ[t,1]
extends to a bounded operator when h is absolutelycontinuous on [0,1) with extra technical conditions aboutwhat happens at 1
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.12
A very nice algebra!
Theorem
(Sarason-Waterman) These operators admit spectralsynthesis. From Sarason’s Banach algebra perspective, thismeans every closed ideal is an intersection of maximal ideals.From Waterman’s perspective, every invariant subspace isspanned by eigenvectors.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.13
Definition (Hilbert Ring)
A Hilbert Ring is a Hilbert space that has a boundedmultiplication defined on it.
Definition (Strictly cyclic algebra)
A commutative strictly cyclic algebra is the set of multipliers{Mx : x ∈ H } where H is a unital commutative Hilbert ring.
Definition (Strictly cyclic operator)
A strictly cyclic operator is a multiplier corresponding to asingly generated unital Hilbert ring.
Definition (Substrictly cyclic operator)
A substrictly cyclic operator is a multiplier corresponding to asingly generated Hilbert ring.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.14
A cool thing...
Being “selfdual", the maximal ideal space of a Hilbert ring livesinside the Hilbert space.
Examples
1 The algebra A of absolutely continuous functions, normedas Sarason did, is a unital Hilbert ring.
2 We can move the multiplicative structure of Sarason’salgebra to L2[0,1] obtaining the multiplication
f ? g = Vf g + f Vg
defined on L2[0,1]
3 Our big basis{χ[0,t] : t ∈ (0,1]
}is then seen to be the
maximal ideal space.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.15
Adjoints of multipliers...
Assume A is a commutative Banach algebra, a ∈ A, and Mathe multiplier on A:
Ma(b) = ab.
1 Every multiplicative functional is an eigenvector for M∗a .2 The eigenspaces are one-dimensional when a is a
generator.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.16
To use quotients...
Definition
A commutative Banach algebra A is Shilov regular when everyclosed subset of the maximal ideal space can be separatedfrom points not in it using elements of A:
< a,e >= 0 for e ∈ E but < a, f >6= 0
1 This is exactly what one needs to say that, for each closedE in the maximal ideal space, the maximal ideal space ofA/E⊥ is E .
2 This is a property lacking in many of the traditionalexamples of strictly cyclic algebras, those arising fromweighted shifts
3 Sarason’s algebra has this property, which is why Kalisch’smethod of restricting his operator to subspaces yielded anoperator with pure point spectrum.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.17
Use tensor products to fatten...
1 If H is Sarason’s Hilbert ring, then its spectrum is [0,1]
2 The Hilbert tensor product is also a Hilbert ring (that canbe identified with L2(I)) whose spectrum is the unit square.
3 Shilov regularity persists
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.18
Recapturing Kalisch...
1 Let T = Mx + V , a generator of Sarason’s algebra withspectrum [0,1]
2 The operator A = I ⊗ T + i(T ⊗ I) has spectrum equal tothe unit square.
3 Given a desired compact set, scale it and translate to fitinside the square, call the result E
4 The image of A in the quotient has spectrum E (regularityis used here).
5 The adjoint of this image is (unitarily equivalent to)Kalisch’s restriction!
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.19
Many new examples of strictly cyclic algebras andoperators...
Theorem
Given any compact subset of the plane, there exists a rationallystrictly cyclic operator whose spectrum equals that compactset.
Theorem
Given any polynomially convex compact subset of the plane,there exists a strictly cyclic operator whose spectrum equalsthat compact set.
Theorem
Given any compact subset of Euclidean space, there exists acommutative semisimple strictly cyclic algebra whose spectrumequals that compact set.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.20
Concept of substrictly cyclic operator...
Examples
1 Mx + V is a generator relative to Sarason’s multiplication
f ? g = Vf g + f Vg
2 Any Hilbert-Schmidt diagonal operator with distinct entries,the multiplier corresponding to a generator for pointwisemultiplication on `2
(ai)(bi) = (aibi)
3 The Volterra operator is also an example, with convolutionmultipication
f ◦ g(x) =
∫ x
0f (s)g(x − s)ds
Every substrictly cyclic operator is the restriction of a strictlycyclic operator to a maximal ideal.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.21
Can recapture another Theorem of Sarason:
Theorem
1 The strongly closed algebra generated by the Volterraoperator is maximal abelian.
2 A Kaplansky density result holds: the operators in the unitball of the strongly closed algebra generated by theVolterra operator are strong limits of operators in the unitball of multipliers.
3 The identity element is in the strongly closed algebragenerated by just the Volterra operator.
The ultra simple proof: there is an approximate identity inL2[0,1] for convolution, and the corresponding multipliers arecontractions.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.22
Can use this theory to characterize the strongly closedideals in the Volterra algebra
Theorem
1 The strongly closed ideals form a continuous chain It witht ∈ (0,1).
2 The annihilator of It is I1−t .3 These ideals consist entirely of nilpotents: the ideal I1/2
consists of square zero nilpotents.4 The ideal I1/n consists of nilpotents of order n.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.23
Examples
• The multiplication is on `2 via
(ai)(bi) = (aibi)
• The strictly cyclic algebra is
αI +
0 0 0 0 0
x1 x1 0 0 0x2 0 x2 0 0x3 0 0 x3 0
... 0 0. . . 0
• For the substrictly cyclic algebra, the multipliers are the
diagonal Hilbert-Schmidt operators, and the substrictlycyclic algebra is the algebra of all bounded diagonaloperators.
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.24
Big Bases
Ben Mathes
Overview
KalischOne dimensional example
Two dimensional example
Sarason - WatermanSarason
Waterman
Their result
Strictly cyclic algebrasSarason’s algebra
Tensor products
New Examples
Substrictly cyclicalgebrasSarason, Erdos, ...
Ideals
a substrict algebra
End——–.25
Dedicated to Heydar Radjavi
