# FOM: #86:Invariant Subspace Problem/fA not= U

Harvey Friedman friedman at math.ohio-state.edu
Wed Mar 29 09:37:19 EST 2000

```In #85, I carefully formulated the general frameworks for

1. Equational Boolean Relation Theory.
2. Inequational Boolean Relation Theory.
3. Conditional Boolean Relation Theory.

4. Simple Compound Equational Boolean Relation Theory.
5. Simple Compound Inequational Boolean Relation Theory.
6. Simple Compound Conditional Boolean Relation Theory.

7. Compound Equational Boolean Relation Theory.
8. Compound inequational Boolean Relation Theory.
9. Compound Conditional Boolean Relation Theory.

These theories are developed in what I called Boolean Relation Theory
contexts (BRT contexts), which are triples (V,K,U), where

i) V is a set of functions;
ii) U is a set;
iii) K is a set of subsets of U.

I have been talking about how the Invariant Subspace Problem (for Hilbert
space) is a very special case of Boolean Relation Theory, ever since Rich
Laver said this in a phone conversation. I was thrilled with his remark,
which suggests the elegant unifying power of Boolean Relation Theory.

postings Tue, 28 Mar 2000 22:32, Wed, 29 Mar 2000 00:16, and #85, Wed, 29
Mar 2000 00:58.

I now want to carefully restate the situation.

INVARIANT SUBSPACE PROBLEM

It says the following:

Every bounded linear operator on Hilbert space has a nontrivial closed
invariant subspace. I.e.,

Every bounded linear operator on Hilbert space has a closed invariant
subspace that is not {0} and not Hilbert space.

INVARIANT SUBSPACE PROBLEM AS EQATIONAL BOOLEAN RELATION THEORY

Let V = the set of all bounded linear operators on Hilbert space H.
Let K = the set of all nontrivial closed subspaces of H; i.e., not {0} and
not H.
Let U = H.

The invariant subspace problem then asserts:

For all f in V there exists A in K such that fA containedin A.

INVARIANT SUBSPACE PROBLEM AS INEQUATIONAL BOOLEAN RELATION THEORY

This allows us to replace K with

K' = the set of all closed subspaces of H.

The invariant subspace problem then asserts:

For all f in V there exists A,B in K' such that fA containedin A, fB
containedin B, A not= B, A not= H, B not= H.

THE INEQUATION fA not= U

Recall the section "some undergraudate exercises in Boolean relation
theory" in posting Tue, 28 Mar 2000 22:32. This concerned the status of the
statements

"For all f:E into E there exists a proper subset A of E of the same
cardinality as E such that fA containedin A"
"For all f:E^2 into E there exists a proper subset A of E of the same
cardinality as E such that fA containedin Q"
"For all multivariate f from E into E there exists a proper subset A of E
of the same cardinality as E such that fA containedin A"

for various sets E.

Now consider:

1) "For all f:E into E there exists a proper subset A of E of the same
cardinality as E such that fA not= E."
2) "For all f:E^2 into E there exists a proper subset A of E of the same
cardinality as E such that fA not= E."
3) "For all multivariate f from E into E there exists a proper subset A of
E of the same cardinality as E such that fA not= E."

We have already discussed the very interesting case where E is countably
infinite. Then 1) is provable in RCA_0, 2) is provable in ACA_0, 3) is not
provable in ACA_0 but is provable in RCA_0 plus the classical infinite
Ramsey theorem. The latter result suggests that 3) cannot be proved without
using the classical infinite Ramsey theorem.

Here is another undergraduate exercise in Boolean relation theory:

THEOREM. 1) holds if and only if E is infinite. If the cardinality of E is
a weakly compact cardinal then 3) holds.

I'll stop here.

**********

This is the 86th in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones are:

1:Foundational Completeness   11/3/97, 10:13AM, 10:26AM.
2:Axioms  11/6/97.
3:Simplicity  11/14/97 10:10AM.
4:Simplicity  11/14/97  4:25PM
5:Constructions  11/15/97  5:24PM
6:Undefinability/Nonstandard Models   11/16/97  12:04AM
7.Undefinability/Nonstandard Models   11/17/97  12:31AM
8.Schemes 11/17/97    12:30AM
9:Nonstandard Arithmetic 11/18/97  11:53AM
10:Pathology   12/8/97   12:37AM
11:F.O.M. & Math Logic  12/14/97 5:47AM
12:Finite trees/large cardinals  3/11/98  11:36AM
13:Min recursion/Provably recursive functions  3/20/98  4:45AM
14:New characterizations of the provable ordinals  4/8/98  2:09AM
14':Errata  4/8/98  9:48AM
15:Structural Independence results and provable ordinals  4/16/98
10:53PM
16:Logical Equations, etc.  4/17/98  1:25PM
16':Errata  4/28/98  10:28AM
17:Very Strong Borel statements  4/26/98  8:06PM
18:Binary Functions and Large Cardinals  4/30/98  12:03PM
19:Long Sequences  7/31/98  9:42AM
20:Proof Theoretic Degrees  8/2/98  9:37PM
21:Long Sequences/Update  10/13/98  3:18AM
22:Finite Trees/Impredicativity  10/20/98  10:13AM
23:Q-Systems and Proof Theoretic Ordinals  11/6/98  3:01AM
24:Predicatively Unfeasible Integers  11/10/98  10:44PM
25:Long Walks  11/16/98  7:05AM
26:Optimized functions/Large Cardinals  1/13/99  12:53PM
27:Finite Trees/Impredicativity:Sketches  1/13/99  12:54PM
28:Optimized Functions/Large Cardinals:more  1/27/99  4:37AM
28':Restatement  1/28/99  5:49AM
29:Large Cardinals/where are we? I  2/22/99  6:11AM
30:Large Cardinals/where are we? II  2/23/99  6:15AM
31:First Free Sets/Large Cardinals  2/27/99  1:43AM
32:Greedy Constructions/Large Cardinals  3/2/99  11:21PM
33:A Variant  3/4/99  1:52PM
34:Walks in N^k  3/7/99  1:43PM
35:Special AE Sentences  3/18/99  4:56AM
35':Restatement  3/21/99  2:20PM
38:Existential Properties of Numerical Functions  3/26/99  2:21PM
39:Large Cardinals/synthesis  4/7/99  11:43AM
40:Enormous Integers in Algebraic Geometry  5/17/99 11:07AM
41:Strong Philosophical Indiscernibles
42:Mythical Trees  5/25/99  5:11PM
43:More Enormous Integers/AlgGeom  5/25/99  6:00PM
44:Indiscernible Primes  5/27/99  12:53 PM
45:Result #1/Program A  7/14/99  11:07AM
46:Tamism  7/14/99  11:25AM
47:Subalgebras/Reverse Math  7/14/99  11:36AM
48:Continuous Embeddings/Reverse Mathematics  7/15/99  12:24PM
49:Ulm Theory/Reverse Mathematics  7/17/99  3:21PM
50:Enormous Integers/Number Theory  7/17/99  11:39PN
51:Enormous Integers/Plane Geometry  7/18/99  3:16PM
52:Cardinals and Cones  7/18/99  3:33PM
53:Free Sets/Reverse Math  7/19/99  2:11PM
54:Recursion Theory/Dynamics 7/22/99 9:28PM
55:Term Rewriting/Proof Theory 8/27/99 3:00PM
56:Consistency of Algebra/Geometry  8/27/99  3:01PM
57:Fixpoints/Summation/Large Cardinals  9/10/99  3:47AM
57':Restatement  9/11/99  7:06AM
58:Program A/Conjectures  9/12/99  1:03AM
59:Restricted summation:Pi-0-1 sentences  9/17/99  10:41AM
60:Program A/Results  9/17/99  1:32PM
61:Finitist proofs of conservation  9/29/99  11:52AM
62:Approximate fixed points revisited  10/11/99  1:35AM
63:Disjoint Covers/Large Cardinals  10/11/99  1:36AM
64:Finite Posets/Large Cardinals  10/11/99  1:37AM
65:Simplicity of Axioms/Conjectures  10/19/99  9:54AM
66:PA/an approach  10/21/99  8:02PM
67:Nested Min Recursion/Large Cardinals  10/25/99  8:00AM
69:Baby Real Analysis  11/1/99  6:59AM
70:Efficient Formulas and Schemes  11/1/99  1:46PM
71:Ackerman/Algebraic Geometry/1  12/10/99  1:52PM
72:New finite forms/large cardinals  12/12/99  6:11AM
73:Hilbert's program wide open?  12/20/99  8:28PM
74:Reverse arithmetic beginnings  12/22/99  8:33AM
75:Finite Reverse Mathematics  12/28/99  1:21PM
76: Finite set theories  12/28/99  1:28PM
77:Missing axiom/atonement  1/4/00  3:51PM
79.Axioms for geometry  1/10/00  12:08PM
80.Boolean Relation Theory  3/10/00  9:41AM
81:Finite Distribution  3/13/00  1:44AM
82:Simplified Boolean Relation Theory  3/15/00  9:23AM
83: Tame Boolean Relation Theory  3/20/00  2:19AM
84: BRT/First Major Classification  3/27/00  4:04AM
85:General Framework/BRT   3/29/00  12:58AM

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